Jonathan Sasson

Case Western Reserve University, Cleveland, Ohio

Small Scale Mass Flow Plug Calibration

NASA/CR—2015-218820

July 2015
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Jonathan Sasson

Case Western Reserve University, Cleveland, Ohio

Small Scale Mass Flow Plug Calibration

NASA/CR—2015-218820

July 2015

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

I would like to extend my gratitude to my adviser, Dr. Paul Barnhart, for his support and guidance throughout the course of my 

thesis work. His expertise was invaluable. I am a better engineer because of his dedication as an adviser. I want to thank my 

committee members, Dr. Joseph Prahl and Dr. Yasuhiro Kamotani for their comments and suggestions that improved the quality 

of my thesis. I also want to thank Dr. David Davis for allowing me to test at NASA Glenn Research Center (GRC) and getting the 

funding to do so. Without his efforts this work could not have been accomplished. He helped procure funding from the LERCIP 

intern program for the numerical study, and from the High-Speed Project of NASA’s Fundamental Aeronautics Program for the 

experimental work. Lastly, I thank Sierra Lobo for allowing me to take the necessary time to complete my work and supporting 

me through the last year of my education.

Available from

Level of Review: This material has been technically reviewed by NASA technical management.

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.

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This report is available in electronic form at http://www.sti.nasa.gov/ and http://ntrs.nasa.gov/

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.
Contents

List of Figures i

List of Tables iii

List of Symbols iv

List of Acronyms vii

Acknowledgments viii

Abstract ix

1 Introduction 1

1.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Analysis Methodology 6

2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Minimum Geometric Area . . . . . . . . . . . . . . . . . . . . 9

2.2 Control Volume Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Inviscid Solution . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Skin Friction Model . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Coupled System/Viscous Model . . . . . . . . . . . . . . . . . 18

2.2.4 Boundary Layer Analysis . . . . . . . . . . . . . . . . . . . . . 21

2.2.4.1 Momentum Integral - Waltz’ Method . . . . . . . . . 21

2.2.4.2 Waltz’ Method Solution . . . . . . . . . . . . . . . . 24

2.2.4.3 Method of Solution . . . . . . . . . . . . . . . . . . . 25

2.3 CFD Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Calibration Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1.1 Turbulence Model . . . . . . . . . . . . . . . . . . . 29

2.3.1.2 Stability Condition . . . . . . . . . . . . . . . . . . . 29

2.3.1.3 Convergence . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2 Distortion Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 30

NASA/CR—2015-218820 iii
3 Experiment 32

3.1 6WB Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.2 LVDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 ASME Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.1 Baseline Calibration . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2 Distortion Generation . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Results and Discussion 49

4.1 Non-Distorted Calibration . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Distortion Description . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Pattern Descriptors . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Numerical Descriptors . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Pressure Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 Non-Distorted Case . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.2 Distorted Case . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Distortion Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Conclusion 76

Appendix A Method of Solution 79

A.0.1 Single Varibale Newton-Raphson Method . . . . . . . . . . . . 79

A.0.2 Multi-Variable Newton-Raphson Method . . . . . . . . . . . . 80

A.0.3 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 82

A.0.4 Fourth Order Runge-Kutta . . . . . . . . . . . . . . . . . . . . 84

A.0.5 Bicubic Spline Interpolation . . . . . . . . . . . . . . . . . . . 85

Appendix B Uncertainty Analysis 87

Appendix C Engineering Drawings 89

Bibliography 102

NASA/CR—2015-218820 iv
List of Figures

1.1 Low-boom inlet test at NASA GRC 8x6ft SWT. . . . . . . . . . . . . 1

1.2 Low-boom inlet/MFP configuration. . . . . . . . . . . . . . . . . . . 2

2.1 Components of the calibration rig and control volume elements. . . . 7

2.2 Schematic for calculating MFP minimum area. . . . . . . . . . . . . . 10

2.3 Minimum geometric area . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Sample control volume with the flow quantities. . . . . . . . . . . . . 14

2.5 Main Components of the N+2 Calibration Rig. . . . . . . . . . . . . 26

2.6 Mesh of N+2 calibration rig including the plenum. . . . . . . . . . . . 28

3.1 15 by 15 cm SWT facility without experimental hardware. . . . . . . 32

3.2 Test setup and instrumentation stations. . . . . . . . . . . . . . . . . 35

3.3 Upstream view of the AIP instrumentation rake. . . . . . . . . . . . . 38

3.4 Upstream view of the MFP instrumentation rake. . . . . . . . . . . . 39

3.5 Mass-flow plug in housing. . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Distortion devices modeled after measured AIP profile. . . . . . . . . 47

4.1 MFP discharge coefficient results. . . . . . . . . . . . . . . . . . . . . 52

4.2 ASME nozzle calibration. . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 MFP discharge coefficient results with reduced ”ideal” mass flow rate. 54

4.4 Numerical calibration results at different pressure ratios. . . . . . . . 55

4.5 Circumferential distortion. . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Ring circumferential distortion. . . . . . . . . . . . . . . . . . . . . . 60

4.7 Pressure variations in non-distorted calibration. . . . . . . . . . . . . 63

4.8 AIP versus MFP distortion. . . . . . . . . . . . . . . . . . . . . . . . 65

4.9 Inlet and CFD distortion results. . . . . . . . . . . . . . . . . . . . . 69

4.10 Experimental distortion results for screen 15.1. . . . . . . . . . . . . . 70

4.11 Distortion results for all screens. . . . . . . . . . . . . . . . . . . . . . 71

4.12 Distortion results for screen 7. . . . . . . . . . . . . . . . . . . . . . . 72

4.13 Distortion results for screen 8.1. . . . . . . . . . . . . . . . . . . . . . 72

4.14 Distortion results for screen 9.1. . . . . . . . . . . . . . . . . . . . . . 73

4.15 Distortion results for screen 14.1. . . . . . . . . . . . . . . . . . . . . 73

4.16 Distortion results for screen 15.1. . . . . . . . . . . . . . . . . . . . . 74

4.17 Distortion results with weighted flow corrected by MFP pressure. . . 75

C.1 MFP calibration rig assembly. . . . . . . . . . . . . . . . . . . . . . . 90

NASA/CR—2015-218820 v
C.2 MFP support assembly- part 1. . . . . . . . . . . . . . . . . . . . . . 91

C.3 MFP support assembly- part 2. . . . . . . . . . . . . . . . . . . . . . 92

C.4 MFP support assembly- part 3. . . . . . . . . . . . . . . . . . . . . . 93

C.5 Settling chamber- part 1. . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.6 Settling chamber- part 2. . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.7 Settling chamber- part 3. . . . . . . . . . . . . . . . . . . . . . . . . . 96

C.8 Connecting flange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.9 ASME nozzle- part 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C.10 ASME nozzle- part 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

C.11 ASME nozzle- part 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

NASA/CR—2015-218820 vi
List of Tables

3.1 Test Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Distortion screens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Summary of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 48

NASA/CR—2015-218820 vii
List of Symbols

A Area

AR Area ratio

C Courant-Friedrichs-Lewy number

CD Discharge coefficient

Cf Skin friction coefficient

C0 Sutherland’s law temperature constant

D Diameter

Fc van Driest’s parameter

Fi Algebraic functions in Walz’s method

h Height

H Shape factor

KA2 Radial and circumferential distortion descriptor

Kr Radial distortion descriptor

Kθ Circumferential distortion descriptor

L Length

m Slope linear equation

m ̇ Mass flow rate

M Mach number

N Integer

p Coordinate point

P Pressure

P r Prandtl number; pressure ratio

q Dynamic pressure

s Slant height

S Surface area

r Radius; recovery factor

R Specific gas constant

Re Reynolds number

t Time

T Temperature

U Velocity

x, y, z Cartesian coordinates

r, θ Cylindrical coordinates

w Corrected flow

W Shape factor

NASA/CR—218820 viii
Z Momentum thickness parameter

Greek Symbols

β Dimensionless parameter in Walz’s method

γ Specific heat ratio

δ Velocity boundary layer thickness; referred pressure

δ

∗ Displacement thickness

δ3 Dissipation thickness

 Numerical tolerance

μ Viscosity

ν Kinematic viscosity

φ Cone angle

θ Polar coordinate angle; momentum thickness; referred temperature

ρ Density

τ Shear stress

Ω Square root of Fc

Subscripts

0 Initial or reference value

1 Upstream or first value

2 Downstream or second value

actual Actual

aw Adiabatic wall

c Contour

calc Calculated value from control volume method

d Distorted

D Diameter

e Exit value; boundary layer edge

eff Effective

ideal Ideal

L Length

min Minimum

max Maximum

plug Plug’s position s

Surface

turb Turbulent

T Total (stagnation) condition

u Undistorted

w Wall

NASA/CR—2015-218820 ix
Superscripts

+ Law of the wall

() Average quantity

n Exponential constant

0 Difference in plug position

NASA/CR—2015-218820 x
List of Acronyms

AIP Aerodynamic Interface Plane

ASME American Society of Mechanical Engineers

CAD Computer Aided Design

CFD Computational Fluid Dynamics

CFL Courant Friedichs Lewy

CSV Comma Separated Variables

CV Control Volume

ESP Electronically Scanned Pressure

GRC Glenn Research Center

LVDT Linear Variable Differential Transducer

MFP Mass Flow Plug

MPR Multiple Per Revolution

QEVC Quiet Experimental Validation Concept

SST Shear Stress Transport

SWT Supersonic Wind Tunnel

NASA/CR—2015-218820 xi

A simple control volume model has been developed to calculate the discharge coefficient through a mass flow plug (MFP) and validated with a calibration experiment.

The maximum error of the model in the operating region of the MFP is 0.54%. The

model uses the MFP geometry and operating pressure and temperature to couple continuity, momentum, energy, an equation of state, and wall shear. Effects of boundary

layer growth and the reduction in cross-sectional flow area are calculated using an integral method. A CFD calibration is shown to be of lower accuracy with a maximum

error of 1.35%, and slower by a factor of 100. Effects of total pressure distortion are

taken into account in the experiment. Distortion creates a loss in flow rate and can

be characterized by two different distortion descriptors.

Small Scale Mass Flow Plug Calibration 

Jonathan Sasson 

Case Western Reserve University 

Cleveland, Ohio 44106 

Abstract 

NASA/CR—2015-218820 xiii
Chapter 1

Introduction

Figure 1.1: Low-boom inlet test at NASA GRC 8x6ft SWT.

Prior to coupling a high-speed inlet to an actual turbine engine, the performance

of the inlet is usually evaluated by testing the inlet with a device that simulates

the air flow demand of the engine. The inlet must also hold stable operation over a

range of mass flow rates. Accurate measurement of the mass flow rate is therefore

NASA/CR—2015-218820 1
CHAPTER 1. INTRODUCTION

Figure 1.2: Low-boom inlet/MFP configuration.

necessary. Due to its mechanical simplicity, the air flow through an inlet is often

controlled and measured with a mass flow plug which is typically a pipe whose exit

area is regulated by translating a body of revolution along the axis of the pipe [1].

The body of revolution is usually conical, but other shapes have been used, and the

pipe may be constant area with a sharp corner or variable area with a contour at

the exit. Similar mass flow plugs are also used during the testing phase to control

and measure boundary-layer bleed flow typically incorporated in high-speed inlets for

shock wave stability and enhanced performance.

To achieve the highest mass-flow measurement accuracy, the mass flow plug must

be calibrated prior to testing. Most often, calibration of the mass flow plug is

performed with well-defined, uniform flow conditions. During inlet testing shockwave/boundary-layer interactions and flow separation in the subsonic diffuser can

significantly distort the flow approaching the mass flow plug. To mitigate the effects

of distortion on the mass flow plug calibration, the usual practice of providing at

least 10 diameters of development length ahead of the mass flow plug (MFP) is recommended. On larger scale inlets models or in applications where there are multiple

engines, 10 diameters are not always practical so the effects of flow distortion must

be accounted for during the calibration.

The consequences of distortion on intake testing are apparent in a recent test

NASA/CR—2015-218820 2
CHAPTER 1. INTRODUCTION

performed by Boeing where they characterized the flow through an inlet integrated

onto a low-boom supersonic cruise aircraft concept in NASA GRC’s 8x6ft Supersonic

Wind Tunnel (SWT). Figure 1.1 shows a photograph of the Boeing Quiet Experimental Validation Concept (QEVC) model installed in the 8x6ft SWT with the MFP

prominently featured on the starboard side. The MFP was of the type shown in

Figure 1.2 [2] and was close-coupled to the inlet as shown. In this figure, the rakes

at the Aerodynamic Interface Plane (AIP) station characterize the flow that would

enter the engine and the rakes at the MFP station are used for calibration of the

MFP. Prior to the 8x6ft SWT test, Boeing experimentally calibrated the MFP under

undistorted flow conditions. During the 8x6ft SWT inlet test, the flow conditions,

with various levels of flow distortion, were recorded at the AIP and MFP rake stations. The levels of total pressure distortion were high enough to question whether

the MFP calibration could be used reliably to determine the flow rate through the

inlet.

1.1 Calibration

The ultimate goal of the calibration is to determine a discharge coefficient, CD, Equation 1.1, for the mass flow plug over an operating range typical of the supersonic testing inlet.It is defined as the ratio of actual mass flow rate to ideal mass flow rate. In

other words, it is the ratio of the mass flow rate at the discharge end of a flow device

to that of an idealized device expanding at the same initial and final conditions.

CD =

m ̇ actual

m ̇ ideal

(1.1)

where  ̇mideal is the ideal inviscid flow rate through the mass flow plug.

The difference between theoretical and actual values is due to viscous effects of the

NASA/CR—2015-218820 3
CHAPTER 1. INTRODUCTION

boundary layer on the wall and heat addition. Apart from friction forces viscosity

also creates a boundary layer. The fluid in the boundary layer is moving slower

compared to the core-flow and reduces the effective flow area. A meaningful measure

of the boundary layer thickness is the displacement thickness. It is a measure of the

distance by which the internal potential flow is displaced inwards as a consequence

of the decrease in velocity in the boundary layer.

Varying the plug position creates a means to vary the back pressure on the inlet.

The position is varied by an electric motor and it is measured by a Linear Variable

Differential Transducer (LVDT). The four primary components of the calibration rig

are the American Society of Mechanical Engineers (ASME) inlet flow nozzle assembly,

an adapter spool piece, the cold-pipe assembly, and the mass-flow plug assembly. The

ASME nozzle measures the flow rate, and draws air from either a plenum tank or the

atmosphere. The wind tunnel exhausters are used as a vacuum source to establish

flow through the assembly and to assure the flow is choked at the MFP throat. Plug

position and minimum area are defined differently for each plug.

1.2 Distortion

Distortion characterizes the non-uniformity of flow at entry to the engine. It is typically measured in terms of the variation of total pressure across the face of the engine

– the aerodynamic interface plane. This simple descriptor does not take into account

radial or azimuthal variations in pressure. Failure to understand the full nature of

the flow field and the sensitivity of the engine to distortion has led to performance

and stability problems [3].

Ultimately, it is the engine sensitivity which determines whether the total pressure

distribution is a sufficient descriptor of the intake distortion. The same criteria can

be applied to MFP testing of an inlet. Although great efforts are taken to avoid

NASA/CR—2015-218820 4
CHAPTER 1. INTRODUCTION

distortion during MFP calibration, some level of it is always apparent during actual

testing. Significant distortion approaching the MFP results in an increase in the level

of uncertainty in the indicated inlet mass flow rates. To reduce the level of uncertainty,

the MFP should be also calibrated with distorted flow; this leads to a distortion

descriptor that correlates the discrepancies between undistorted and distorted flow.

NASA/CR—2015-218820 5
Chapter 2

Analysis Methodology

2.1 System Description

Due to the high costs of wind tunnel testing, it is often not feasible to calibrate the

flow nozzle or the MFP. Recently, there has been a substantial increase in the demand

of CFD predictions since these are cheaper. They are still time consuming and require

the researcher to work closely with a CFD analyst. The goal of this analysis is to

create a one-dimensional model that can be implemented by a researcher for any

calibration geometry, and obtain a solution that is of comparable accuracy as the

CFD results.

A rapid method for calculating the discharge coefficient, CD, for a Mass Flow

Plug (MFP) is desired that can be used in lieu of an experimentally determined

calibration. The method must be able to capture the geometric variations of typical

mass flow plugs operating under a range of flow rates, pressures, and temperatures.

A control volume analysis has been developed to perform this calculation of the

discharge coefficient for nominal, undistorted, axisymmetric flows through a MFP.

The control volume analysis developed for this work is comprised of a sequence

of flow calculations through the MFP. The first pass uses an inviscid, compressible

NASA/CR—2015-218820 6
CHAPTER 2. ANALYSIS METHODOLOGY

Figure 2.1: Components of the calibration rig and control volume elements.

NASA/CR—2015-218820 7
CHAPTER 2. ANALYSIS METHODOLOGY

one-dimensional stream-tube model. The second pass calculates a wall shear based on

the results from the first inviscid calculation as the flow properties at the edge of the

boundary layer using a compressible, turbulent skin friction model. The third pass

solves a coupled set of one-dimensional relations for conservation of mass, momentum,

energy, an equation of state, and the wall shear calculated in the second calculation.

The result of these successive calculations is an accurate one-dimension model of the

velocity, pressure, and temperature through the MFP.

The discharge coefficient calculation must also include the effects of boundary

layer growth, and the reduction in cross-sectional flow area as characterized by the

boundary layer displacement thickness. The last calculation in the sequence uses

an integral method to calculate the growth of the boundary layer, from which the

displacement thickness is then determined.

The goal of the calibration is to establish a relationship between plug position and

discharge coefficient. The calibration rig must be long enough in order for the flow

to settle and prevent total pressure distortion. A baseline calibration is established

without any distortion devices. Transverse variation in total pressure is due solely to

boundary layer growth. This baseline is used to compare against a calibration which

deliberately creates total pressure distortion to better simulate inlet conditions. It is

important to note that some total pressure loss is always present in the baseline calibration due to wall friction and boundary layer growth. This loss must be accounted

for to establish a more accurate trend between the baseline and distorted calibration.

Figure 2.1 shows the basic physical elements of the calibration rig. Although

different MFPs use slightly different configurations, they all follow the same basic

guidelines. They all include an intake nozzle that measures the flow rate and draws

air from a source (either a plenum tank or the atmosphere), a cold pipe with pressure

rakes, and the MFP. Locations of quantities of interest are also shown in Figure 2.1.

The flow is assumed choked at the MFP throat, therefore the exit Mach number, Me,

NASA/CR—2015-218820 8
CHAPTER 2. ANALYSIS METHODOLOGY

is 1. The ideal mass flow rate is determined by the intake total pressure , PT0, the

total temperature, TT0, and the MFP minimum area, Ae. The exit total pressure,

PT e, is calculated by accounting for losses in the rig. This pressure is used to calculate

the actual mass flow rate.

The calibration process is modeled by a series of control volumes across the calibration rig as shown in Figure 2.2. The analysis is also split into four sequential

steps. The first step calculates the thermodynamic properties across each control

volume isentropically. The second step uses the previous solutions to calculate the

wall friction losses. The third step ties together the losses with the isentropic solution

using a fully-coupled system of equations to determine the total pressure loss. Finally,

the fourth step computes the displacement thickness to correct the effective flow area

reduced by boundary layer displacement. The results from these four steps are used

to calculate the discharge coefficient, CD.

2.1.1 Minimum Geometric Area

The minimum geometric area (or throat area) of the MFP must be determined to

start the first pass of the control volume analysis. The sonic line is assumed to be

perpendicular to surface, therefore the minimum geometric area is approximated by

a curve, f(x), that is both normal to the cone and the cold pipe’s exit as shown in

Figure 2.2 and revolved around the x-axis. Since the cold pipe has a contoured exit

(with a radius, rc, of 0.43 inches), the location of the curve that creates the minimum

area varies by plug position. From Figure 2.2 it is deduced that the minimum area

lies between the two tangent points of the contour: between 45◦and 110◦. This is

shown by the white circles on the contour.

To start the calculation a coordinate system with its origin at the tip of the cone

is established. The x location of the center of the contour’s circle is given by:

NASA/CR—2015-218820 9
CHAPTER 2. ANALYSIS METHODOLOGY

Figure 2.2: Schematic for calculating MFP minimum area.

xc = L − x

0

plug (2.1)

The point (x2, y2) at which f(x) intercepts the contoured exit is defined from the

center of the contour and by the angle θ2.

x2 = rc cos θ + xc

y2 = h − rc sin θ (2.2)

where h (1.177 inches) is the height from the center line to the center of the

contour. A second angle, theta1, is used to define the point at which f(x) intercepts

the cone. Since the cone is at an angle of 45 degrees, x1 = y1. Therefore:

x1 =

h + xc tan θ1

1 + tan θ1

The curve f(x) is defined by satisfying four distinct boundary conditions. These

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boundary conditions are used to find the four constants that define the curve as a

cubic. It must pass through both (x1, y1) and (x2, y2) and be perpendicular to both

surfaces. Perpendicularity to the cone is satisfied by imposing the surface of the cone

as the normal line of f(x) at (x1, y1). The radius of curvature of f(x) at (x2, y2)

must equal the contour’s radius, rc to satisfy perpendicularity. The four boundary

conditions are defined as:

f(x1) = y1

f(x2) = y2

f

0

(x1) = −1

rc =











[1 + f

0

(x2)

2

]

3/2

f

00(x2)











(2.3)

With the four constants known, the area is solved for by revolving the curve

around the a-axis. The surface area of a revolution is determined by

Athroat = 2π

Z x2

x1

f(x)

p

1 + f

0

(x)

2dx (2.4)

Equation 2.4 must be solved for numerically using the 3/8 Simpson’s rule. The

minimum area depends on both θ1 and θ2. The area is optimized by varying the

angles, and recomputing the area until the minimum area is reached. The surface

area of the revolution is represented in Figure 2.3 as the darker surface.

2.2 Control Volume Analysis

To determine the discharge coefficient as given in Equation 1.1 it is necessary to first

determine the ideal mass flow rate. A one-dimensional compressible mass flow rate

relation is required to account for compressibility effects in the MFP and calibration

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Figure 2.3: Minimum geometric area

rig. The mass flow rate is determined as a function of area, A, Mach number, M, total

pressure, PT , total temperature, TT , and gas properties by assuming air is an ideal

gas with constant specific heat. The resulting compressible mass flow rate function

and its development are presented in Equations 2.5 through 2.11 [4].

m ̇ = ρUA (2.5)

U = M

p

γRT (2.6)

ρ =

P

RT

(2.7)

m ̇ = A

P

√

T

r

γ

R

M (2.8)

P = PT



1 +

γ − 1

2

M2

−

γ

γ−1

(2.9)

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T = TT



1 +

γ − 1

2

M2

−1

(2.10)

m ̇ = A

PT

√

TT

r

γ

R

M



1 +

γ − 1

2

M2

−

γ+1

2(γ−1)

(2.11)

In order to determine the internal flow characteristics of the MFP and calibration

rig it is first discretized into a number of segments. The geometry of each section

contributes to create a series of one-dimensional control volumes. The inner surfaces

create the real boundaries, and the entrance and exit cross-sectional areas of each

segment create the imaginary boundaries of the control volume element. The sum

of the volume elements create a stream-tube representation of the internal control

volume [5]. Each control volume element is further subdivided into smaller sections

to create a stream-tube from very small discretized control volume sections. Figure

2.4 represents a visualization of a one-dimensional discretized control volume element.

The figure also shows the stream-tube the control-volumes create. The internal flow

solutions of the one-dimensional stream-tube correspond to physical locations on the

calibration rig (shown in Figure 2.1).

The required geometric information for each one-dimensional control volume element of Figure 2.4 consists of the upstream and downstream areas, A1 and A2

respectively, and the total surface area, S. All flow quantities are assumed constant

perpendicular to the flow across the volume elements, and are only calculated at the

upstream and downstream boundaries. The flow quantities calculated at each imaginary boundary of the control volume are: density, ρ, axial velocity, U, pressure, P,

temperature, T, total temperature, TT , total pressure, PT , Mach number, M.

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Figure 2.4: Sample control volume with the flow quantities.

2.2.1 Inviscid Solution

The first step in the methodology is to compute the flow quantities presented in

the previous section from a simple statement of conservation of mass presented in

Equation 2.8.

m ̇ 1 =  ̇m2 (2.12)

Equation 2.11 and 2.12 are combined to develop compressible one-dimensional

stream tube relation, and it is shown in Equation 2.13. Even though Equation 2.13

takes into account changes in total pressure and temperature, the flow is modeled

isentropically in the first step. Therefore the upstream total pressure and temperature

remain constant along the stream tube. The only parameter that corresponds to a

change in Mach number is the change in flow area across a volume element.

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CHAPTER 2. ANALYSIS METHODOLOGY

A1

PT1

√

TT1

r

γ

R

M1



1 +

γ − 1

2

M2

1

−

γ+1

2(γ−1)

= A2

PT2

√

TT2

r

γ

R

M2



1 +

γ − 1

2

M2

2

−

γ+1

2(γ−1)

(2.13)

The first pass analysis starts at the bellmouth’s entrance and marches downstream

across each volume element, using Equation 2.13 to determine the downstream Mach

number from the known upstream Mach number and cross-sectional areas. The downstream Mach number then becomes the upstream Mach number for the next volume

element marching the solution downstream. The temperatures and pressures are determined from the known Mach numbers and Equations 2.9 and 2.10 respectively.

Velocity and density are computed from Equations 2.6 and 2.7. Due to the isentropic

assumption, the total pressure and temperature are constant across each volume element.

Equation 2.13 cannot be solved explicitly for the downstream Mach number M2.

It is therefore solved iteratively using a single variable Newton-Raphson iteration

scheme. Equation 2.13 is rewritten as Equation 2.14 such that each term on the

left-hand-side is normalized and approximately unity in magnitude. This helps the

convergence of the iteration scheme. The ratio of total pressure and temperature is

exactly 1 under inviscid assumptions and therefore not shown in Equation 2.14.



A1

A2



−

M2

M1

"

1 + γ−1

2 M2

2

1 + γ−1

2 M2

1

#−

γ+1

2(γ−1)

= 0 (2.14)

To start the first step in the analysis, Equation 2.14 is used to solve for the

bellmouth’s entrance Mach number. The upstream and downstream areas correspond

to the bellmouth’s entrance and MFP minimum geometric exit, respectively. The

downstream Mach number is 1 since the MFP is choked at the exit for the pressure

ratio imposed. Equation 2.14 is solved using the single variable Newton-Raphson

iteration scheme.

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2.2.2 Skin Friction Model

The inviscid flow solution determined in the first step is used to calculate the surface

shear force using boundary layer theory. The inviscid velocity, density, temperature,

and Mach number become the boundary layer edge quantities, denoted by the subscript e. The wall shear per unit surface area is given in Equation 2.15 as a function of

density and velocity at the boundary layer edge, and the skin friction coefficient, Cf .

Since the edge velocity and density are known from the previous inviscid calculation,

the only variable left to be determined is the skin friction coefficient.

τw =

1

2

ρeU

2

e Cf (2.15)

The flow is assumed turbulent throughout the calculation. Since the pressure

gradient is larger across the throat region and it covers a small segment compared to

the rest of the rig, the calculation assumes no significant pressure gradient. With these

assumptions, a flat plate compressible turbulent skin friction coefficient formulation

of White and Christoph is used. This correlation has proven very accurate over the

entire practical range of Reynolds numbers, Mach numbers, and wall temperatures.

A complete derivation of the method is found in Reference [6].

The simple formula for turbulent skin friction on a flat plate as determined by

White and Christoph is:

Cf ≈

0.455

Ω2ln2



0.06

Ω

ReL

2

μe

μw

q Te

Tw

 (2.16)

Ω is the square root of van Driest’s parameter Fc. The details of this parameter

are also presented in Reference [6].

Ω =

qTaw

Te

− 1

sin−1a

3/2 + sin−1 1

2

√

a

(2.17)

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CHAPTER 2. ANALYSIS METHODOLOGY

where

a =

r

γ − 1

2

M2

e

Te

Taw

(2.18)

The equations presented here are a simplified version from those found in Reference

num by assuming adiabatic flow. Equation 2.16 also accounts for the axisymmetric

geometry of the calibration rig by using the turbulent cone rule, or an approximate

Mangler transformation. This is apparent in the factor of 1/2 in the Reynolds number. The adiabatic wall temperature, Taw, depends on the static temperature, Mach

number and recovery factor,r, as presented in Equation 2.19:

Taw = Te



1 + r

γ − 1

2

M2

e



(2.19)

The recovery factor is calculated by breaking the law of the wall into three different

parts: a linear layer for y

+ < 5, a buffer layer for 5 < y+ < 30, and a log layer

for y

+ > 30. By assuming a constant turbulent Prandtl number, the velocity and

temperature profiles are integrated to obtain an estimate for the turbulent recovery

factor. The details of the derivation are presented in Reference [7].

rturb ≈ P r1/3(2.20)

The predominant term in the calculation of the skin friction coefficient is the

Reynolds number based on length. The length used to calculate the Reynolds number

is not a physical length but rather is defined by Equation 2.23. It is determined by

dimensional arguments. Since the volume element does not account for an axial

length, an effective length is approximated by the surface area, S, of the volume

element. The running length is the sum of all the effective lengths of the preceding

volume elements.

Each control element is modeled as a conical frustum as seen in Figure 2.4. The

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surface area of the frustum is defined by Equation 2.21, where the base and the top

radius of the frustum are defined by R1 and R2 respectively. These can be rewritten

in terms of the cross-sectional areas at the entrance and exit (A1 and A2) of the

volume element shown in Equation 2.22. Substituting Equation 2.22 into Equation

2.21 yields the effective length L shown in Equation 2.23.

S = π(R1 + R2)

p

(R1 − R2)

2 + L2

(2.21)

Rn =

r

An

π

(2.22)

L =

1

√

π

X

N

n=2

p

S2

n − (An − An−1)

2

√

An−1 +

√

An

(2.23)

ReL =

ρeUeL

μe

(2.24)

Finally, the viscosity is calculated by the Sutherland-law approximation and it is

presented in Equation 2.25. The variables μ0 (3.584 ×10−7lb-s/f t2), T0 (491.6 R),

and C0 (198.6 R) are constants in the Sutherland-law approximation specific for the

gas to be modeled [6].

μ(T) ≈ μ0



T

T0

3/2

T0 + C0

T + C0

(2.25)

2.2.3 Coupled System/Viscous Model

The results of the second step do not contain the necessary information to calculate

the discharge coefficient. The effects of wall shear calculated in the second step violate

the assumptions of the isentropic flow model used in the first step. A third step is

required to couple the effects of wall shear stress on the internal MFP calibration rig

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CHAPTER 2. ANALYSIS METHODOLOGY

flow solutions. Wall shear stress creates a small loss in total pressure which results in

a reduction in mass flow compared to the one calculated by isentropic assumptions

in the first step.

The 7 downstream quantities (ρ2, U2,T2, P2, TT2, PT2, and M2) are determined

from the 7 known upstream quantities (ρ1, U1,T1, P1, TT1, PT1, and M1) including

the effects of wall shear [5]. Figure 2.2 is a representation of this system. The 7

necessary governing equations to solve for the 7 unknowns are presented in Equations

2.26 through 2.32. These relations create a system of equations that must be solved

simultaneously, and a multi-variable Newton-Raphson iteration scheme is used. Each

term in the equations are once again normalized to provide a value close to unity to

aid in the convergence of the iteration scheme. The isentropic flow solution is used as

the initial guess for the 7 variables the iteration technique. The 7 equations represent

characteristics every physical control volume consists of [4].

1. Equation of state for an ideal gas:

ρ2

ρ1

P1

P2

T2

T1

− 1 = 0 (2.26)

2. Conservation of mass across the volume element:

ρ2U2A2

ρ1U1A1

− 1 = 0 (2.27)

3. Conservation of momentum across the volume element:

ρ1U1U2A1 − ρ1U

2

1A1 + P2A2 − P1A1 − P(A2 − A1) + τS

(P1 +

1

2

ρ1U

2

1

)A1

= 0 (2.28)

(P and τ are the arithmetic averages of the upstream and downstream values

across the volume element).

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4. Conservation of energy across the volume element:

1 −

TT2ρ2U2A2

TT1ρ1U1A1

= 0 (2.29)

5. Relation between total and static temperature for an ideal gas:

TT2

TT1

−

T2

T1

"

1 + γ−1

2 M2

2

1 + γ−1

2 M2

1

#

= 0 (2.30)

6. Relation between total and static pressure for an ideal gas:

PT2

PT1

−

P2

P1

"

1 + γ−1

2 M2

2

1 + γ−1

2 M2

1

# γ

γ−1

= 0 (2.31)

7. Relation between velocity and Mach number for an ideal gas:

U2

U1

M1

M2

r

T1

T2

− 1 = 0 (2.32)

The third step in the analysis is similar to the first step in that it computes all

the flow properties marching through the volume elements to determine the internal

flow. The difference here is that the total pressure does not remain constant because

of the wall shear stress effects. Even though total temperature remains constant since

there is no heat addition, it is treated as an unknown in the system of equations for

completion. It also serves as a redundancy in the calculations. If the total temperature

changes, then there is an error in the solution.

This section of the analysis also starts at the bellmouth’s entrance and marches

downstream to the throat. The flow through the throat is choked. Since the downstream Mach number (unity) is known across this volume element, the system is

reduced to a 6 by 6 system. Initially the system is using the ideal mass flow rate

determined in the first step (used to calculate the entrance Mach number). In order

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to conserve mass, the total pressure in the last control volume will want increase. To

avoid the erroneous sudden increase in pressure, the total pressure is assumed constant across the last volume element. This reduces the system of equations on the last

control volume even further to a 5 by 5 system. With the new exit total pressure and

Mach number of 1, a new mass flow rate is determined. The entrance Mach number is

recomputed from this value and the solution marches downstream again. The process

is repeated until the mass flow rate is converged. This process determines the entire

MFP calibration rig flow properties while accounting for friction losses.

2.2.4 Boundary Layer Analysis

The third pass analysis determined the internal flow properties for the calibration

rig, however this is still insufficient information to accurately determine the discharge

coefficient. Although the previous calculations took into consideration friction losses

on total pressure effects, they did not account for the boundary layer displacement

thickness which reduces the effective flow area. The third pass determined mass flow

rate is lower than the first pass inviscid mass flow rate, but it yields an unrealistic

discharge coefficient around 0.99. A fourth step is therefore necessary to determine the

boundary layer growth and the displacement thickness. A von K ́arm ́an momentum

integral equation is applied in this step.

2.2.4.1 Momentum Integral - Waltz’ Method

The simple momentum integral equation presented in Equation 2.33 represents a reliable method for calculating the boundary layer displacement thickness. The equation

applies to both laminar and turbulent flows. For compressible flow it is necessary to

include Mach number effects [8].

dθ

dx +

θ

Ue

(2 + H − M2

e

)

dUe

dx =

1

2

Cf (2.33)

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where

H ≡

δ

∗

θ

(2.34)

is the shape factor of the boundary layer velocity profile, δ is the boundary layer

thickness, δ

∗

is the displacement thickness, and θ is the momentum thickness.

δ

∗

(x) = Z ∞

0



1 −

ρ

ρe

u

Ue



dy (2.35)

Equation 2.33 still has too many unknowns (θ, H ). The skin friction coefficient is

defined by Equation 2.16. Walz’s method is applicable to both laminar and turbulent

boundary layer methods in both low- and high-speed flows [6]. The method introduces

a momentum thickness parameter, Z, and another shape factor, W.

Z = θ



ρeUeθ

μw

n

(2.36)

W =

δ3

θ

where n = 1 for laminar flow and n = 0.268 for turbulent flow, and δ3 is the compressible kinetic energy thickness. The two coupled first order differential Equations

2.37 and 2.38 are obtained by the substitution of Z and W in the momentum and

mechanical energy integral relations. Only the parameters Z and W are necessary in

these equations.

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Momentum : dZ

dx +

F1

Ue

dUe

dx Z = F2 (2.37)

Mechanical energy : dW

dx +

F3

Ue

dUe

dx W =

F4

Z

(2.38)

The four functions Fi are algebraic functions of the Mach number, Reynolds number based on momentum thickness 

Reθ =

ρeUeθ

μw



, and shape factor H = δ

∗/θ.

The formulas are simplified by the intermediate parameters a and b, and are

defined as:

a ≈ 0.0394 (W − 1.515)0.7

b ≈ 1 + 0.88

γ − 1

2

M2

e

(W) (2 − W)

The functions F1 to F4 correlated with turbulent data are given by Walz as:

F1 = 2.268 + 1.268H − M2

e

F2 = 1.268

a

b

F3 = 1 − H + 0.88(γ − 1)M2

e

F4 =

1

b

"

2β



ρeUeθ

μw

0.168

− aW#(2.39)

where

H =

δ

∗

θ

≈ 1 + 1.48(2 − W) + 104(2 − W)

6.7

(2.40)

β =



1 + 0.587(γ − 1)M2

e

1 + 0.44(γ − 1)M2

e

n

(2.41)

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The exponent n in the definition of β is the viscosity-temperature exponent, which

is 0.67 for air. Separation occurs at W = 1.515 for either laminar and turbulent flow,

which is why the parameter a equals zero at that point.

2.2.4.2 Waltz’ Method Solution

The two differential Equations 2.37 and 2.38 are integrated using a 4th Order RungeKutta integration method to solve for Z, and W [9]. The integration is carried

downstream starting from bellmouth entrance.

The integration is marched downstream from the starting point across volume

elements. The initial conditions at the upstream location of the control volume are

used to solve for the downstream values of Z, W, and θ. The downstream values

then become the upstream values, and thus the initial conditions for the next volume

element.

The initial value of H is assumed to be 1.35. This value is used to calculate the

initial value of W (1.77). The value of 1.35 is chosen to represent turbulent transition.

The transition value of H is in the range of 1.3 to 1.4. Using W, a and b are calculated,

and used to find θ. Since the skin friction coefficient Cf is known at the entrance,

it can be approximated by the following relation to find θ. This values serves as an

approximation for the initial value of Z.

Cf ≈ 2

a

b



ρeUeθ

μw

0.268

Once the downstream values of θ, Z, W are determined for the last volume element, δ

∗

, is calculated from Equation 2.35. The MFP’s throat radius is reduced using

δ

∗

, from which an effective throat area is calculated. The ideal mass flow rate through

the calibration rig is determined by using the MFP’s throat area, and the supply total pressure and temperature. Since the flow is choked at this point, Equation 2.7 is

reduced to Equation 2.42.

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CHAPTER 2. ANALYSIS METHODOLOGY

m ̇ ideal = Athroat

PT

√

TT

r

γ

R



γ + 1

2

−

γ+1

2(γ−1)

(2.42)

The viscous mass flow rate is finally computed from Equation 2.43 by correcting

the calculated mass flow rate from the third pass by the ratio of effective flow area to

throat area (minimum geometric area). Finally, the discharge coefficient is determined

from Equation 1.1.

m ̇ actual =  ̇mcalc

Aef f

Athroat

(2.43)

where  ̇mcalc is the mass flow rate calculated in the third pass.

2.2.4.3 Method of Solution

Several numerical methods are implemented in the analysis to determine the discharge coefficient. The single-variable Newton-Raphson method is used to calculate

the entrance Mach number and flow rate of the calibration rig. The multi-variable

Newton-Raphon method together with LU decomposition solve the coupled 7 by 7

system to determine the internal flow loss model. Finally, 4th order Runge-Kutta is

implemented to solve for the displacement thickness. All of these methods require

some initial guess and/or value. The details of these techniques are presented in Appendix A along with the details of the cubic spline interpolation. The interpolation

technique is used in a later section [10].

2.3 CFD Analysis

Due to the asymmetrical nature of distortion, it is not trivial to create an analysis

routine using an axisymmetric, simple control volume method. It is necessary to

account for shearing flows with variations azimuthally. Swirl is another phenomenon

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CHAPTER 2. ANALYSIS METHODOLOGY

that arises. To truly capture the effects of distortion, we resort to CFD to solve for

the internal flow in the calibration rig.

It is important to note that the CFD is not capturing all the details of the flow.

Only the total pressure is measured and used as a boundary condition. None of the

turbulence parameters are known, which are important to truly define the flow. For

this reason only profiles which exhibit no flow separation during testing are used in

this analysis. The goal of the CFD analysis is to predict a discharged coefficient

for the calibration rigs with and without distortion. The results are also used to

determine a distortion parameter that characterizes the effects of distortion on the

inlet test by Boeing and GRC.

Figure 2.5: Main Components of the N+2 Calibration Rig.

The first step in the CFD analysis is to create a numerical calibration to serve as

a baseline to compare all other results against. The resulting mass flow rate from the

numerical calibration is the actual flow rate determined in Section 2.2.4.2. This CFD

calibration is compared against the results of the control volume analysis, and actual

test data. The comparison is presented in Chapter 4.

A simplified view of the the Boeing N+2 MFP setup of the hardware is presented

in Figure 2.5. The initial calibration was performed by Boeing at their facility prior

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CHAPTER 2. ANALYSIS METHODOLOGY

to testing the inlet shown in Figure 1.1 at GRC. The MFP used in this procedure is

a conical plug with a 45 degree angle. The variation of the minimum flow area versus

plug position is shown in Chapter 4.

The tool used in the CFD analysis is WIND-US. It is used to numerically solve the

Euler and Navier-Stokes equations governing fluid mechanics, along with supporting

equation sets governing turbulent and chemically reacting flows. WIND-US has no

geometric input, and since WIND-US uses externally generated computational grids,

all geometric input and capability depend on the grid generator. The grid generator

used in this analysis is Pointwise.

2.3.1 Calibration Mesh

The WIND-US flow simulator provides ample flexibility for both structured and unstructured grids. Grid lines can conform to complex shapes or may pass through

regions not in the flow field. The grid may be divided into zones to conform to the

geometry better, to allow grid embedding (i.e., zones with finer grids in regions of

high gradients like boundary layers), and/or to allow parallel computation. At least

15 grid points are necessary to capture the behavior of the boundary layer.

Only a simple 2-D axisymmetric grid of the geometry is necessary to calculate the

discharge coefficient for undistorted flow. The simplified calibration rig CAD assembly

shown in Figure 2.5 was imported into Pointwise and stripped of all unnecessary

components such as the wall thicknesses and pressure rakes. The pressure rakes are

not modeled. A large plenum is added at the MFP exit (shown in Figure 2.6) to

simulate outflow to atmosphere.

The spacing between grid points near the surfaces of all the meshes used in this

analysis is 0.003 inches. The spacing between points near the symmetry line (shown

in Figure 2.6) is 0.047 inches. The distance between each grid point in every zone

is automatically computed by Pointwise to match these boundaries. Grid spacing on

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CHAPTER 2. ANALYSIS METHODOLOGY

Figure 2.6: Mesh of N+2 calibration rig including the plenum.

each side of a common boundary between zones must be equal to produce a smooth

mesh as shown in Figure 2.6. Finer grid spacing near these surfaces appear as a

shadow at the wall.

Before running WIND-US the boundary condition at each inflow station must be

specified. At inflow station 1, total pressure and temperature are specified. These are

constant along the plane to simulate air begin drawn in from a plenum. Each plug

position is numerically calibrated at 5 inlet pressures. The details of this analysis are

presented in Chapter 4. The Mach number must also be specified; it serves as an

initial guess to start the iteration scheme. A Mach number of 0.05 is specified as the

initial guess of the entrance Mach number. The boundary condition at inflow station

2 creates the plenum. The freestream static pressure is specified to be atmospheric.

A Mach number of 0 will cause the program to run into an error, therefore a total

pressure at the inflow station 2 is calculated from Equation 2.9 with a Mach number

of 0.05 and the freestream static pressure.

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2.3.1.1 Turbulence Model

The turbulence model must also be specified. All simulations are run using the Menter

Shear Stress Transport (SST) turbulence model [SST]. Previous studies have shown

that the SST turbulence model performs well for jet flows [ref]. The SST turbulence

model is a robust two-equation eddy-viscosity turbulence model. The model combines

the k-ω turbulence model and k- turbulence model such that the k-ω is used in the

inner region of the boundary layer, and switches to the k- in the free shear flow.

Both models attempt to provide a general description of turbulence by two transport

equations. The first equation is the turbulence kinetic energy while the second is the

specific rate of dissipation of turbulence kinetic energy [11].

2.3.1.2 Stability Condition

Finally, the program requires a maximum Courant–Friedrichs–Lewy (CFL) number

input. It is a necessary stability condition in the numerical analysis of explicit time

integration schemes. The CFL is defined in Equation 2.44

C = ∆t

Xn

i=1

uxi

∆xi

≤ Cmax (2.44)

where

1. C is the CFL number

2. ∆t is the time step

3. ∆xiis the length step in the i spatial variable. It is not required for the length

step to be the same for each spatial variable (i = 1, . . . , n)

4. uxi is the magnitude of the velocity in the spatial variable i.

Cmax changes depending on the method used to discretized the equations (explicit

or implicit). Cmax is typically 1 for explicit methods [11].

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2.3.1.3 Convergence

Convergence is defined when the solution has reached steady state. It is verified by

checking whether the mass flow rate is constant along each i index. If the difference

in mass flow rates in each i index is less than some specified tolerance, the solution

has converged. The mass flow is integrated at each i index across the j plane (and

k in 3D grids). It is calculated by Equation 2.45 [1]. The final mass flow rate is the

average of flow rates between the downstream tangential edge of the radial nose of

the MFP and the cold pipe entrance. The result is given in Equation 2.46.

m ̇ =

Z

θ

Z

r

ρurdrdθ (2.45)

m ̇ =

P

N

i=1

m ̇ i

N

(2.46)

The mass flow sees a spike at the common boundary of each zone, and some initial

concern required a closer look at this phenomenon. The concern was that it could

cause some delay in convergence, or a false average. This effect was determined to be

insignificant, and further details are presented in Chapter 4.

2.3.2 Distortion Mesh

The goal of the distortion analysis is to characterize the effect of total pressure distortion on mass flow rate. During the test shown in Figure 1.1 and 1.2, Boeing measured

a significant level of total pressure distortion. The total pressure distortion was measured at the AIP. The same distortion profiles are applied as boundary conditions at

the AIP to calculate the mass flow rate. However, the test only produces forty data

points, which is not enough to create a fine mesh consisting of 4,800 grid points. The

data is first spliced by a bicubic interpolation scheme to produce boundary conditions

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CHAPTER 2. ANALYSIS METHODOLOGY

for the entire AIP (these are the inflow boundary conditions).

Since the boundary condition is applied at the AIP, there is no need to include

any geometry upstream of it. Therefore the bellmouth is discarded in the distortion

analysis. The grid is created by making a 2-D grid with the desired plug position, and

then revolving it around the x-axis. A 3-D grid is necessary since the applied total

pressure distortion profile is not symmetrical. The same surface boundary conditions

as in the calibration case are applied at the surface of the mesh.

To reduce computational and convergence time with this mesh, the solution method

employed parallel processing along with three levels of grid sequencing. First, a coarse

grid sequence included only every fourth point in each direction. Second, a medium

grid sequence utilized every other point in each direction. Third, a fine sequence used

every grid point in the mesh.

The freestream static pressure this time is calculated from the measured total

pressure. Once again, a total pressure is calculated with the wall static pressure to

obtain the measured Mach number. The case matrix is found in Chapter 4. Convergence is determined in the same manner as in the calibration case, using Equation

2.46. The supercomputer cluster at NASA Ames was used for all CFD runs.

NASA/CR—2015-218820 31
Chapter 3

Experiment

3.1 6WB Facility

Figure 3.1: 15 by 15 cm SWT facility without experimental hardware.

NASA/CR—2015-218820 32
CHAPTER 3. EXPERIMENT

The objective of the experimental work is to validate the analysis presented in

Chapter 2, by testing Boeing’s MFP assembly. The first test is to perform an undistorted flow calibration for comparison to the Boeing experimental calibration, the

CFD numerical calibration, and the control-volume method calibration. The second

test is to place obstruction devices in the contraction of the calibration rig to create distorted inflow conditions and repeat the calibration procedure. The results are

compared to the CFD study of distortion observed during Boeing’s inlet test.

The experimental work was performed using the W6B altitude exhaust system at

NASA GRC (shown in Figure 3.1). Three test sections are connected downstream of

the nozzle. They are attached to a fixed diffuser which exits to the altitude exhaust

system. The altitude exhaust is controlled by the central air system, and can maintain

a back pressure vacuum of about 1.75 psia. This is controlled by a remotely actuated

facility valve (valve AC040). The facility can handle a maximum flow rate of 25 lb/s.

Data acquisition and recording is via a laboratory standard ESCORT D system. Particular data acquisition capabilities include total and static pressure measurements, temperatures, hot wire anemometry, flow visualization, and laser data

techniques. ESCORT also has the capability of evaluating specific parameters and

displaying them real-time. Data (both channel and calculated variables) are provided

as ASCII comma separated variables (CSV) files [2].

ESP is an off the shelf electronically scanned pressure system. The system consists

of numerous 32 port, rack mounted modules of ±15 psid, located outside the test

chamber. Each module has a check pressure to assure the module is performing

correctly and can be calibrated in place using up to 5 highly accurate calibration

pressures to maintain the overall system accuracy of ±0.05% of range. This calibration

is automatically performed every hour [12].

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CHAPTER 3. EXPERIMENT

3.2 Test Setup

The test article assembly is shown in Figure 3.2. The N+2 MFP and the contraction

assembly were designed and fabricated by Boeing. The remainder of the components

were designed and fabricated specifically for this test. Ambient air is drawn through

the ASME nozzle which serves as the primary reference mass-flow measurement device. Further details on the nozzle are provided in Section 3.4.

In this experiment the flow from the ASME nozzle is dumped directly into a

settling chamber which acts as a plenum. The intent of the plenum chamber was for

the flow through the mixing pipe to settle. The nozzle is directly connected to the

settling chamber via an interface flange. An O-ring sitting in a groove on the nozzle’s

flange creates a seal on the interface flange to prevent and flow leakage. From the

settling chamber, the flow passes through a 22:1 contraction and enters the MFP.

The plenum attaches directly to the bellmouth. The connecting flange also has an

o-ring and connects to the bellmouth with 8 #12 screws. The MFP assembly is held

in place via a support structure specifically designed for this experiment.

The MFP assembly consists of a diffuser with an AIP pressure rake at its 1.5

diameter entrance that expands to 1.994 inches. It also has cold pipe of the same

diameter with an MFP pressure rake and an contoured exit. The MFP rake is 2 inches

from the pipe exit. The 2 inch mass flow plug sits at various positions inside the pipe.

The entire test article assembly is mounted on the top window of W6B test section

#2 as shown in Figure 3.2. After passing through the MFP, the flow exhausts into

the test section which is maintained at approximately 1.75 psia (altitude exhaust).

The resultant pressure ratio is 8.22 ensuring a choked MFP.

Since altitude exhaust is the only service required for the experiment, test section

# 3 is blanked off with a 12 inch blank pipe flange. The maximum possible flow rate

through the test article is approximately 0.6 lb/s for these conditions.

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CHAPTER 3. EXPERIMENT

Figure 3.2: Test setup and instrumentation stations.

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CHAPTER 3. EXPERIMENT

3.3 Instrumentation

3.3.1 Stations

The test article is instrumented with steady state instrumentation. The AIP rake has

limited dynamic instrumentation, but this was not used during testing. The MFP

rake also has four thermocouples. Figure 3.2 shows the instrumentation stations

through the test article. Unless otherwise noted, all steady state pressures will be

recorded with 15 psia ESP modulus. The uncertainty of all pressure measurements

is ± 0.00229 psi [12].

Electronically scanned modules are differential pressure measurement units consisting of an array of silicon pressure sensors which are electronically multiplexed

through an on board multiplexer and instrumentation amplifier. These modules are

widely used to measure surface pressure on models and walls of wind tunnels. The

ESP modules contain 32 measurement channels connected to pressure ports through

small diameter tubing.

Moving from the upstream end to the downstream end, the details of each instrumentation station are as follows:

1. Atmospheric Station

Four ESP modules monitor atmospheric pressure in the testing facility.

2. Nozzle Station

The ASME nozzle is the primary reference mass-flow measurement device.

There are eight equally spaced wall taps located at the throat of the nozzle

to measure static pressure. They are shown on the engineering drawing for the

nozzle in Appendix C. The taps are connected together in groups of two creating four distinct static pressure measurements. These pressures are measured

with four 5.0 psid (referenced to atmosphere) Bell & Howell transducers.

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CHAPTER 3. EXPERIMENT

3. AIP Station

The AIP rake sits at the entrance of the 1.5 inch diameter diffuser. The rake

follows the guidelines established in ARP 1420 to measure total pressure at the

AIP [13]. It represents an engine face plane which is defined by the leading edge

of the most upstream engine strut, vane, or blade row. The station contains

eight equally-spaced rakes with five pitot tubes per rake for a total of 40 total

pressures. The pitot tubes are all area-weighted, i.e. the area covered by each

ring is the same. Figure 3.3 represents the layout of the rake. The figure

shows an upstream view of the rake; it represents the pressure distribution a

compressor ”sees”. The nomenclature used in Figure 3.3 that refers to the

individual pitot tubes (i, j) is per ARP246 [14]. The ring number is defined by

i, and the rake number by j.

In addition, there are eight a wall static pressure tap located azimuthally midway between the AIP rakes. The rakes and static pressure taps are connected

to the ESP modules. Every other rake has one dynamic total pressure transducer for a total of four. These transducers (kulites) are not used during the

experiment. They were used during the original Boeing calibration, and the

8x6 ft inlet test. They were left in place for consistency with the previous tests,

and to help generate distortion.

4. MFP Station

The MFP station records the conditions immediately upstream of the mass flow

plug. This station contains four equally-spaced rakes with three pitot tubes per

rake for a total of 12 total pressures. These are clocked 22.5 degrees counter

clock wise (looking upstream) as to avoid the wakes of the upstream AIP rakes.

The tubes on these of rakes are also positioned radially so as to have equal area

weighting. There are no wall static taps at the MFP station. The layout of the

rake is shown in Figure 3.4 in an upstream view.

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CHAPTER 3. EXPERIMENT

Figure 3.3: Upstream view of the AIP instrumentation rake.

In addition, each rake contains a single type-K thermocouple in between the first

and second pitot tube for a total of four total temperature measurements. The

thermocouples produce a voltage output that can be correlated to the temperature that the thermocouple is measuring. The range for type-K thermocouples

is between 132 R to 2742 R. The error for these thermocouples is ± 0.75% above

492 R (0 ◦C) and ± 0.4% below 492 R.

5. Exhaust Station

The exhaust station consists of four equally spaced static pressures to monitor

the MFP back-pressure and pressure ratio across the MFP.

In total there are 72 steady state pressure readings, and four temperature measurements. The instrumentation takes a reading every second. When data is recorded,

the data is averaged over ten seconds to create a single reading of each measurement

point.

NASA/CR—2015-218820 38
CHAPTER 3. EXPERIMENT

Figure 3.4: Upstream view of the MFP instrumentation rake.

3.3.2 LVDT

The MFP has a diameter of 2 inches and a half cone angle of 45 degrees. The cone

has a 1 inch length and a shoulder of 1.052 inches. The drive system includes a 24

VDC electric motor with two limit switches. The plug is instrumented with an LVDT

which is used to determine the plug position. The jackscrew is 1.75 inches deep into

the shoulder of the plug. The LVDT position is calibrated during model installation

by setting plug at the full closed limit switch position and measuring the distance

between the aft face of the MFP and the jackscrew housing with calipers (Figure 3.5).

The plug will then be translated to the full open limit switch position incrementally

during testing

3.4 ASME Nozzle

The ASME Long-Radius Flow Nozzle is one the most important components of the

testing section as it serves two purposes. It draws the fluid into the settling chamber,

NASA/CR—2015-218820 39
CHAPTER 3. EXPERIMENT

Figure 3.5: Mass-flow plug in housing.

and measures the actual mass flow rate flowing through the MFP. The nozzle generates differential pressure which is used to determine the mass flow rate. The flow

rate is corrected by a nozzle discharge coefficient.

The ASME nozzle’s shapes consists of a quarter ellipse convergence section and a

cylindrical throat section. The ellipse is a function of the throat diameter, DASME.

The major radius is 2/3 of DASME, and the minor radius equals DASME. A throat

diameter of 1.5 inches was determined to limit the throat Mach number, MASME, to

0.8 at the maximum plug position (i.e. fully open). This is to avoid choking the nozzle

and have it limit the mass flow rate instead of the plug. Post manufacturing inspection

revealed that DASME is exactly 1.500 inches. The uncertainty in the measurement is

± 0.0005 inches. The nozzle is manufactured out of aluminum with a surface finish

of 32 micro-inches.

The advantages of the nozzle over the venturi are its shorter length and lower

manufacturing cost. To reduce the permanent head loss, the exit flow does not

exhaust into a sudden expansion but rather into a shallow angle (7.5 ◦) conical diffuser

(recovery cone). In cases operating without a recovery cone, the permanent head loss

has been measured up to 40% of the differential pressure [15]. The shallow angle is

determined by the ellipse that defines the geometry of the nozzle. A flange at end of

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CHAPTER 3. EXPERIMENT

the conical diffuser attaches the nozzle to the rest of the assembly.

With the geometry known, the air flowing through the nozzle can be approximated

as an ideal gas. The wall taps measure the differential pressure and determine the

static pressure at the throat. They are shown in the engineering drawing provided

in Appendix C as well as on the overall system description of Figure 3.2. Using

the atmospheric pressure as the total pressure and the measured static pressure, the

throat Mach number is calculated from Equation 3.1.

MASME =

vuut

2

γ − 1

 

PT0

P

γ−1

γ

− 1

!

(3.1)

The final parameter needed to calculate the flow rate is the nozzle discharge

coefficient. It takes into account any loses due to viscous effects, and permanent

head loss. The nozzle is calibrated in the flow lab at GRC to determine a correlation

for discharge coefficient based on the throat Reynolds number. The calibration curve

and the Reynolds number calculation are presented in Chapter 4. Finally, the massflow rate is determined from the Mach number and discharge coefficient where AASME

is the throat cross-sectional geometric area.

m ̇ = CDAASME

PT0

√

TT0

r

γ

R

MASME 1 +

γ − 1

2

M2

ASME−

γ+1

2(γ−1)

(3.2)

3.5 Operation

The maximum pressure ratio (8.22) is used during testing to ensure a hard choked

MFP. Thus a single run consists of simply translating the MFP over an axial translation range specified later in this section. A sample of the full test matrix involves

a calibration sweep with uniform flow conditions, followed by a number of runs with

various distortion generation devices. Before testing any distortion devices, a baseline

calibration without any distortion generators is established. The test matrix is seen

NASA/CR—2015-218820 41
CHAPTER 3. EXPERIMENT

in Table 3.1.

3.5.1 Baseline Calibration

The calibration begins by positioning the plug 95% of the way forward from the

jackscrew housing. The plug is not closed all the way to avoid damaging the MFP

or the cold pipe. The back pressure is then dropped to approximately 1.75 psia.

The plug is opened by increments of 5% of the closed plug position to measure the

mass flow rate at various plug positions. The Mach number at the AIP is monitored

during testing and the plug is opened until the Mach number is approximately 0.7 to

simulate realistic engine operation. At each plug position the pressure is first allowed

to settle for a few seconds before a data sample is recorded. Each sampling records

the data over 10 seconds and averages the data to obtain a single reading for each

instrument.

3.5.2 Distortion Generation

Distortion is created by a ring, rapidly prototyped out of ABS plastic, that fits in the

throat of the bellmouth contraction 3 inches above the AIP. Each ring has a mesh

that partially covers different sections of the AIP. The mesh wire has a thickness

and spacing of 0.125 inches. The settling chamber is lifted with a crane to insert

the distortion generator. Each device is tested at three different orientations: 0, 22.5

and 337.5 degrees. The orientation is such that at 0 degrees lines with AIP rake

#1. The goal is to create high pressure areas the MFP rakes will be able to capture.

Several devices are tested to obtain levels of distortion at the MFP similar to those

of the CFD study. The plug position is swept starting from the same position as in

the calibration run, however it is stopped when the AIP Mach number is about 0.6.

There are two reasons for this lower Mach number; to match the Mach numbers seen

in the CFD study and to avoid breaking the mesh and damaging the pressure rakes.

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CHAPTER 3. EXPERIMENT

Date Test Config. Pressure

Ratio

MFP

PositionsDuration Notes

09/16/14 1 D0-1 Max 14 15 min. Baseline Calibration

09/16/14 2 D1-1 Max 9 10 min. Device #2, at 0◦

09/16/14 3 D1-1 Max 9 10 min. Device #2, at 22.5◦

09/16/14 4 D1-1 Max 9 10 min. Device #2, at 337.5◦

09/16/14 5 D1-2 Max 9 10 min. Device #3, at 0◦

09/16/14 6 D1-2 Max 9 10 min. Device #3, at 22.5◦

09/16/14 7 D1-2 Max 9 10 min. Device #3, at 337.5◦

09/16/14 8 D1-3 Max 9 10 min. Device #8, at 0◦

09/16/14 9 D1-4 Max 9 10 min. Device #7, at 337.5◦

09/17/14 10 D0-2 Max 14 15 min. Baseline Calibration

09/17/14 11 D1-4 Max 9 10 min. Device #7, at 0◦

09/17/14 12 D1-4 Max 9 10 min. Device #7, at 22.5◦

09/17/14 13 D1-5 Max 9 10 min. Device #8.1, at 0◦

09/17/14 14 D1-5 Max 9 10 min. Device #8.1, at 22.5◦

09/17/14 14 D1-5 Max 9 10 min. Device #8.1, at 337.5◦

09/17/14 16 D1-6 Max 9 10 min. Device #7.1, at 0◦

09/17/14 17 D1-6 Max 9 10 min. Device #7.2, at 337.5◦

09/17/14 18 D1-7 Max 9 10 min. Device #8.2, at 0◦

09/17/14 19 D1-7 Max 9 10 min. Device #8.2, at 22.5◦

09/17/14 20 D1-7 Max 9 10 min. Device #8.2, at 337.5◦

09/19/14 21 D0-3 Max 14 15 min. Baseline Calibration

09/19/14 22 D1-8 Max 9 10 min. Device #9, at 0◦

09/19/14 23 D1-8 Max 9 10 min. Device #9, at 22.5◦

09/19/14 24 D1-8 Max 9 10 min. Device #9, at 337.5◦

09/19/14 25 D1-9 Max 9 10 min. Device #9.1, at 0◦

09/19/14 26 D1-9 Max 9 10 min. Device #9.1, at 22.5◦

09/19/14 27 D1-9 Max 9 10 min. Device #9.1, at 337.5◦

09/19/14 28 D1-10 Max 9 10 min. Device #12, at 337.5◦

09/19/14 29 D1-11 Max 9 10 min. Device #7.2, at 337.5◦

10/01/14 30 D0-4 Max 14 15 min. Baseline Calibration

10/01/14 31 D1-12 Max 9 10 min. Device #15.1, at 0◦

10/01/14 32 D1-12 Max 9 10 min. Device #15.1, at 22.5◦

10/01/14 33 D1-12 Max 9 10 min. Device #15.1, at 337.5◦

10/01/14 34 D1-13 Max 9 10 min. Device #14.1, at 0◦

10/01/14 35 D1-13 Max 9 10 min. Device #14.1, at 22.5◦

10/01/14 36 D1-13 Max 9 10 min. Device #14.1, at 337.5◦

10/01/14 37 D0-5 Max 19 20 min. Baseline Calibration

Table 3.1: Test Matrix

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CHAPTER 3. EXPERIMENT

Value The rings with just a mesh create low levels of MFP distortion. To create the

higher levels observed in the Boeing inlet test, parts of the mesh are blocked with

duct tape. The idea is to create a low pressure area and a jet on the opposite side

of the ring. The mesh is very thin, and at high speeds it may break and damage

the pressure rakes. Duct tape makes the mesh more flexible. Two distortion rings

are created that match the high and low pressure regions of AIP distortion observed

during the inlet test. One of these rings is shown in Figure 3.6.

Screen #

Max. MFP

Distortion

Screen

2 6.21%

3 6.34%

7 17.72%

7.1 16.21%

NASA/CR—2015-218820 44
CHAPTER 3. EXPERIMENT

7.2 19.62%

8 10.67%

8.1 21.76%

8.2 0.01

9 18.08%

9.1 28.37%

NASA/CR—2015-218820 45
CHAPTER 3. EXPERIMENT

12 5.15%

14.1 39.61%

15.1 37.09%

Table 3.2: Distortion screens.

3.6 Uncertainties

Table 3.3 shows a summary of the uncertainties of the various measurement devices.

All of these lead to an uncertainty in the calculated and ideal mass flow rate, and

discharge coefficient. There is also some uncertainty in the ASME nozzle discharge

coefficient. This is by far the largest error and has the most effect on the error

calculations presented in Appendix B. The error for various points in this discharge

coefficient can be seen in Figure 4.2.

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CHAPTER 3. EXPERIMENT

Figure 3.6: Distortion devices modeled after measured AIP profile.

NASA/CR—2015-218820 47
CHAPTER 3. EXPERIMENT

Source Value

ASME throat diameter 0.0005 in

ESP pressure reading 0.0029 psi

Differential pressure on ASME

nozzle 0.005 psi

Thermocouples 0.001 R

Plug position 0.001 in

Table 3.3: Summary of uncertainties

NASA/CR—2015-218820 48
Chapter 4

Results and Discussion

4.1 Non-Distorted Calibration

The experiment was performed five times without any distortion devices to produce

a baseline calibration using non-distorted flow. The results of this calibration are

presented in Figure 4.1 showing the MFP’s discharge coefficient, CD, and the MFP

area ratio, AR. The area ratio is defined as the throat area determined in Chapter 2

over the cold pipe’s maximum cross-sectional area.

AR =

Athroat

Amax

(4.1)

The experimental results are compared against predictions made by the control

volume method presented in Chapter 2 and a CFD calibration. The ideal mass flow

rate is calculated from the total upstream conditions and the throat area for all

solutions. The region of interest in the calibration lies between an area ratio of 0.3

and 0.5. It is in this region that the MFP is used as a flow meter during wind tunnel

testing. In this region the maximum deviation of the control volume calculations

from the experimental results is 0.54%. The maximum deviation in the CFD results

is 1.35%. The maximum experimental error in the operating range is 0.57%. The

NASA/CR—2015-218820 49
CHAPTER 4. RESULTS AND DISCUSSION

locations of these errors are shown in Figure 4.1 by the small arrows. The error

analysis is presented in Appendix B. The control volume method is within the error

bounds of the experimental data. Both the control volume method and the CFD

analysis are within the experimental error in the operating range.

The simple control volume analysis developed from first principals has a higher

accuracy than the CFD solution calibration by approximately 0.8%. Figure 4.1 shows

the control volume method is a reliable alternative to calibrate a MFP assembly.

It also reduces the pre-computational time since no grid is required, only a simple

description of the physical geometry. Each CFD solution took about 8 hours to

converge using the supercomputer clusters as NASA Ames, while the control volume

method took 5 minutes to calculate the calibration curve over the range of MFP area

ratios presented.

The actual experimental flow rate is calculated from the ASME nozzle as shown

in Chapter 3. The nozzle’s discharge coefficient is determined from a metrology

calibration. The calibration is presented in Figure 4.2 showing the nozzle’s discharge

coefficient and the Reynolds number at the throat, ReASME. The Reynolds number

is computed by the following steps:

1. Static temperature at the throat is determined from Equation 2.10 using the

throat’s Mach number, MASME (determined in Chapter 3), and the total temperature.

2. Density is computed from the static pressure and temperature at the ASME

nozzle’s throat using the ideal gas law.

3. Velocity is calculated from the throat’s Mach number and the static temperature: VASME =

√

γRTMASME.

4. Viscosity is calculated from the static temperature using Sutherland’s law (presented in Chapter 2).

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CHAPTER 4. RESULTS AND DISCUSSION

5. Reynolds number is determined from the parameters determined in 2 through

4 and the throat diameter.

The metrology results show a significant amount of scatter, with a maximum

deviation of 1.52% at ReASME = 100, 000, and do not cover the entire range tested

during the MFP calibration. The discharge coefficient has a sudden drop for Reynolds

numbers lower than 50,000, however no test data is taken below 65,000. A least

squares fit is used to determine the discharge coefficient between the minimum and

the maximum tested Reynolds number. This curves matches the typical curve of

an ASME nozzle drawing air from the atmosphere as shown by the dashed curve in

Figure 4.2. The typical wind tunnel test operating range lies in the region calibrated

in the metrology lab. The scatter could account for the increased error in the control

volume method outside the operating range in Figure 4.1. The maximum error in the

control volume method is 1.79%. This maximum error is near the plug’s minimum

limit.

Because wind tunnel testing engineers at NASA use the total pressure measured by

the MFP rakes instead of the total upstream pressure to calculate a reduced “ideal”

flow rate to calculate the discharge coefficient. This discharge coefficient does not

account for any viscous losses upstream of the plug. The results of calculating the

discharge coefficient with this reduced ideal mass flow rate are presented in Figure

4.3. The figure also includes calibration results performed by Boeing who use the

MFP pressures to correct their calibration. The behavior of the curves shown in

Figure 4.3 is vastly different than in Figure 4.1. Viscous effects are not prevalent in

the operating range, which is not the expected behavior with increasing mass flow

rate. Boeing’s calibration is within the experimental error of the new calibration.

The CFD results were calculated for pressure ratios of 1.4 to 5 to create choked

and unchoked conditions at the throat. The pressure ratio, P r, is defined as the

upstream total pressure over the exit back pressure (static). The upstream pressure

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CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.1: MFP discharge coefficient results.

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CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.2: ASME nozzle calibration.

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CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.3: MFP discharge coefficient results with reduced ”ideal” mass flow rate.

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CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.4: Numerical calibration results at different pressure ratios.

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CHAPTER 4. RESULTS AND DISCUSSION

in the CFD calibration was varied from 14.7 to 75 psia compared to 14.35 psia during

the test. The results presented in Figure 4.4 are from a simulation at 75 psia and

pressure ratio of 5. These results are compared against the calibration performed

by Boeing at the same pressure ratios. The curves represent the weighted mass flow

rates over the range of pressure ratios at the MFP station for various area ratios.

Corrected flow rate, w, is the mass flow rate that passes through a device if the

inlet pressure and temperature correspond to ambient conditions at sea level on a

standard day. The flow is corrected by the pressure and temperature observed at the

MFP (θ = TT /518.7 and δ = PT /14.696). There is a minute negative slope to the

line due to the slight increase in viscosity with pressure. This result is negligible is

does not affect the results significantly.

w =  ̇m

√

θ

δ

(4.2)

The best agreement between the CFD and Boeing results occurs after a pressure

ratio of 2, where the MFP is hard choked. The maximum error in this region is 1.78%.

The CFD results are not as accurate for lower pressure ratios which correspond to

the transonic region. The CFD overestimates the flow in this region, and the error

grows with increasing flow rate (area ratio).

4.2 Distortion Description

4.2.1 Pattern Descriptors

Distortion is traditionally visualized by plotting total pressure contours at the AIP

in polar coordinates (r,θ). The study the the flow behavior as it travels downstream

contours are also plotted at the MFP station. Visual patterns are useful to relate

the total pressure defect regions to the source of problems in the intake flow. The

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CHAPTER 4. RESULTS AND DISCUSSION

experimental and numerical results are presented as pressure contours throughout

this chapter. They are always upstream looking, meaning it is the total pressure distribution the engine “sees” approaching the compressor blades. The black diamonds

in all of the contours represent the position of each pitot tube probe at the AIP, and

MFP station. These are created by cubic spline interpolation of the test data. The

pressure is normalized by the upstream total pressure.

4.2.2 Numerical Descriptors

Numerical parameters define and correlate the loss of engine performance and stability

due to distortion. These parameters describe the radial and circumferential intensities

and extents of distortion in quantified terms. For each radial ring of probes, four

elements are typically defined as recommended in ARP 1420 [16]:

1. The circumferential intensity element

2. The circumferential extent element

3. The multiple-per-revolution element

4. The radial intensity element

The circumferential intensity (∆P C/P) element refers to the amplitude of a distortion pattern. It is defined such that the total pressure defect at each radial position

is the difference between the average pressure at that radius, or the ring average pressure (PT,i), and the average minimum pressure at that radius (PTlow,i ), normalized

by the ring average pressure. Positive values of intensity indicate a region where the

pressure lies below the face average pressure.

(∆P C/P)i =

PT,i − PTlow,i

PT,i

(4.3)

NASA/CR—2015-218820 57
CHAPTER 4. RESULTS AND DISCUSSION

PT,i =

1

2π

Z

2π

0

P(θ)i, dθ (4.4)

PTlow,i =

1

θ

−

i

Z

θ2i

θ1i

P(θ)i, dθ (4.5)

P(θi) is determined from linear fit of the data points in each ring.

The distortion extent as described by ARP 1420 refers to the circumferential arc

size of a distorted region. The circumferential extent for each ring is the angular

region, in degrees, in which the pressure is below ring average pressure.

θ

−

i = θ2i − θ1i (4.6)

The total pressure distortion for the fifth ring of the pattern shown in Figure 3.7

is shown below in Figure 4.6. The graph also shows the low pressure region and the

average low pressure of the ring. This pattern is described as a one-per-rev pattern

(meaning MPR = 1) since it only has one region where the pressure is below the

average ring pressure.

The multiple-per-revolution element is a correction term and describes the number

of equivalent low pressure regions for each ring. The equivalence is based on the ratio

of the total integrated area beneath the average pressure line. It is defined (for Q

low pressure regions) by summing the product of the intensity and the extent of each

probe and dividing by the maximum product.

(MP R)i =

X

Q

k=1

∆P C

P



ik

θ

−

ik

∆P C

P



ik

θ

−

ik

MAX

(4.7)

NASA/CR—2015-218820 58
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.5: Circumferential distortion.

The radial intensity element is defined by the difference between the face average

total pressure at the AIP (PT ) and the ring average pressure, normalized by the AIP

average total pressure.



∆P R

P



i

=

PT − PT,i

PT

(4.8)

The average total pressure at the AIP is the area-weighted value to be consistent

with industry standards and recommendations given in ARP 1420. A positive value

represents a ring average pressure that is below the face average.

PT =

1

N

X

N

i=1

PT,i (4.9)

The above equation assumes that the N radial rings of probes are located in areaweighted positions. Figure 4.7 is an illustration of the radial distortion of the pattern

shown in Figure 3.7.

NASA/CR—2015-218820 59
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.6: Ring circumferential distortion.

The distortion intensity elements may be combined to form overall distortion parameters. The resulting parameters are then used to screen inlet distortion data

against engine distortion tolerance limits. The required algebra to for some parameters can be very complicated and in some cases expensive to process.

A common descriptor that takes into account both radial and circumferential total

pressure distortion is the KA2 descriptor [16]. It is defined as follows

KA2 = Kθ + Kr (4.10)

where Kθ is the circumferential distortion and Kr is the radial distortion.

Kθ =

X

N

i







1

ri







 

P

M

j

PT cos θ

!2

+

 

P

M

j

PT sin θ

!2







1/2







q

X

N

i



1

ri



(4.11)

NASA/CR—2015-218820 60
CHAPTER 4. RESULTS AND DISCUSSION

Kr =

X

N

i

 PT

r

2.8

i

∆P R

P



i



q

X

N

i



1

r

2.8

i



(4.12)

where q, the average dynamic pressure, is given by

q =

γ

2

P M

2

(4.13)

4.3 Pressure Decay

4.3.1 Non-Distorted Case

Although no distortion devices were used during the baseline calibration, some small

amount of total pressure distortion was measured by the AIP and MFP probes. The

distortion levels are defined by the difference of the maximum and minimum pressure

divided by the average pressure at the corresponding face. This simple descriptor is

typically used during inlet testing.

DIST =

PT,max − PT,min

PT

(4.14)

The AIP distortion is mostly radial and noticeable every other rake, where the

kulites are located. The kulite blockage creates a small low pressure region around

the AIP probes. The rest of the flow is unaffected by them. This is shown in Figure

4.7(a) for the non-distorted case with the maximum pressure variation. Total pressure

distortion grows downstream and is increased at the MFP due to viscous effects.

These effects are evident in Figure 4.7(b) where the total pressure has small radial

variations at the MFP station. The small amount of distortion due to the kulites

settles downstream and it is not very noticeable at the MFP. Both the AIP and MFP

NASA/CR—2015-218820 61
CHAPTER 4. RESULTS AND DISCUSSION

pressure readings are interpolated azimuthally to produce smooth contours.

The results presented in Figure 4.7(c) show the MFP distortion levels increasing

linearly as AIP distortion increases. This corresponds to an increase in area ratio as

the plug is opened. The CFD results do not detect any significant pressure variations

at the AIP. Although the levels at the MFP are higher than the AIP, they are still not

comparable to those measured experimentally for a given area ratio. The maximum

error in the CFD results is 90%. The CFD results are not reliable in measuring total

pressure variations at a given face.

4.3.2 Distorted Case

The variation between distortion at the AIP and MFP is presented in Figure 4.8.

The CFD distortion is measured by using only the 12 points that correspond to the

probes’ locations at the MFP station. Experimental distortion tends to decay more

rapidly than the CFD would suggest. The CFD solution has a shallower, slope and it

is not able to accurately predict how well the flow settles downstream. The AIP rakes,

which are not modeled in the CFD study, act as crude screens effectively lowering

the distortion levels. Increased turbulent mixing in the experiment also lowers the

distortion levels. The CFD has no other information on the flow other than the total

pressure at the AIP; it does not reflect the turbulence structure, streamwise gradients

in the flow nor capture secondary flow component that might be present during inlet

testing. The maximum deviation from the experimental values in the CFD results is

33%.

Figure 4.8 also shows the actual MFP distortion levels measured during the inlet test. The CFD model is still not able to accurately predict how well the flow

settles compared against the inlet results. The maximum deviation from the inlet

test results is lower compared to the new experimental values (16.67%), it is still not

a good approximation. These results also include low distortion levels at the AIP

NASA/CR—2015-218820 62
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.7: Pressure variations in non-distorted calibration.

NASA/CR—2015-218820 63
CHAPTER 4. RESULTS AND DISCUSSION

not experienced during the inlet test. These levels are similar to those experienced

during the test run under non-distorted conditions. In this region the viscous effects

dominate and the distortion grows down the MFP.

4.4 Distortion Results

To study the effects of total pressure distortion on the MFP calibration a total of

13 different devices were tested to obstruct the flow at the AIP. The experimental

results are compared against CFD results. The CFD study consists of 35 different

total pressure patterns at the AIP observed during the previous inlet test. The 35

patterns vary over 11 different plug positions. The experiment covers the range of

plug positions used for the CFD results. As mentioned previously in Chapter 2, the

measured total pressure distribution is used as the upstream boundary condition.

Each case takes an average of 50 hours to converge using grid sequencing and parallel

processing.

The CFD results are presented in Figure 4.9 as total pressure contours at the AIP

and MFP face. Figure 4.9(a) is the pressure profile at the AIP used as a boundary

condition. The 40 total pressures and 8 static pressures are used to create 4,800 points

for a fine grid. The average of the inner probe ring is used as the center pressure to

create the inner region not measured during testing. Figure 4.9(b) is the measured

profile at the MFP station during the inlet test. Figure 4.9(c) is the CFD result at the

MFP station using all 4,800 points, while Figure 4.9(d) represents a profile created by

extrapolating the pressures at the MFP probe locations and interpolating the resuts.

From (c) it is evident that the cold pipe is not long enough for the flow to settle..

The distortion levels at (b) and (d) are 24.86% and 27.87% respectively. By comparing

(c) to (d) it is clear that the MFP rake profile is too coarse and misses a large section

of the high pressure region. The MFP rake cannot capture enough points in the high

NASA/CR—2015-218820 64
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.8: AIP versus MFP distortion.

NASA/CR—2015-218820 65
CHAPTER 4. RESULTS AND DISCUSSION

pressure region to have a similar distortion levels compared to the full CFD solution.

The distortion in (c) is 34.95%.

The result of the MFP rake being too coarse is also observed during the experiment

where the same distortion device was tested at three distinct circumferential locations.

Although the distortion at the AIP only varied by 5%, distortion at the MFP varied

as much as 15% in some cases by changing the orientation of the device at the same

plug position. Various devices were tested, each with a different amount of blockage.

Higher distortion levels were observed by leaving an open area and blocking another

section, creating a high and low pressure region.

Figure 4.10 shows the results of using distortion device #15.1 to recreate a similar

distortion pattern observed in Figure 4.9. The low pressure region measured in the

experiment is much larger and not comparable to the inlet results. Screen 15.1 is not

a good representation of the inlet distortion pattern. Distortion at the AIP varies

between 55.58% and 60.57% depending on the orientation. The MFP distortion varies

between 22.87% and 37.12%. Both the CFD and inlet test results are in this range

of distortion.

The distortion results or all screens are shown in Figure 4.11. The y-axis is

the difference between distorted and undistorted discharge coefficient divided by the

undistorted discharge coefficient, CD,u.

∆CD = CD,d − CD,u (4.15)

The discharge coefficient is determined by using the upstream pressure to calculate

the ideal mass flow rate. The results are compared against the CFD results. For the

same range of distortion, the CFD under predicts the mass flow losses.

Although the experimental and CFD trend-lines have a different y-intercept, they

follow the same trend of decreasing discharge coefficient as distortion increases. Distortion is a source of pressure loss and reduces the mass flow rate. There is a significant

NASA/CR—2015-218820 66
CHAPTER 4. RESULTS AND DISCUSSION

amount of scatter in the results shown in Figure 4.11. The distorted behavior depends

on the screen used to produce it. The scatter is reduced by looking at each screen

individually. It is necessary to take into account the radial and circumferential effects

of distortion to classify distortion.

Figures 4.12 through 4.16 represent the results for 5 individual screens and compared to their corresponding CFD solution when applicable. Not all screens match a

certain CFD profile. Each figure represents a different mesh at three distinct orientations (0◦

, 22.5

◦ and 337.5◦

). The results show that the distorted discharge coefficient

is dependent not only on plug position, but also on the individual screens. Each

screen has a different distortion pattern which may be defined by the KA2 and DIST

descriptors. The results can be used to correct MFP results measured under distortion conditions but calibrated with undistorted flow, such as during Boeing’s inlet

test.

NASA typically presents distortion results by correcting the flow by the arithmetic

average of the total pressure at the MFP station. The pitot tubes are positioned in

such a way that the average is area-weighted. The result of this correction is shown in

Figure 4.17. The y-axis shows the difference in weighted flow between the distorted,

wd, and undistorted cases, wu.

∆w = wd − wu (4.16)

The difference is normalized by the undistorted weighted flow. Both the experimental and CFD results show a significant amount of scatter around the trend-line.

The maximum experimental error in the weighted flow is 0.85%, therefore the trendline is not very useful. The maximum deviation of the experimental results to the

trend-line is 38.35%. Using the trend-line to correct inlet tests is not reliable because

of this scatter. Correcting the mass flow rate by the MFP total pressure shows an

increase in flow rate, which contradicts the experimental results shown in Figure 4.9.

NASA/CR—2015-218820 67
CHAPTER 4. RESULTS AND DISCUSSION

The MFP is extremely sensitive to distortion levels. The MFP presented in this chapter is a poor design. The straight length segment of the cold pipe is not long enough

to allow the flow to settle. Typical MFP configurations use a straight length segment

of at least 12 plug diameters to avoid distortion. The straight length segment in this

MFP is 3 diameters.

NASA/CR—2015-218820 68
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.9: Inlet and CFD distortion results.

NASA/CR—2015-218820 69
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.10: Experimental distortion results for screen 15.1.

NASA/CR—2015-218820 70
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.11: Distortion results for all screens.

NASA/CR—2015-218820 71
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.12: Distortion results for screen 7.

Figure 4.13: Distortion results for screen 8.1.

NASA/CR—2015-218820 72
CHAPTER 4. RESULTS AND DISCUSSION 73

Figure 4.14: Distortion results for screen 9.1.

Figure 4.15: Distortion results for screen 14.1.

NASA/CR—2015-218820 73
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.16: Distortion results for screen 15.1.

NASA/CR—2015-218820 74
CHAPTER 4. RESULTS AND DISCUSSION

Figure 4.17: Distortion results with weighted flow corrected by MFP pressure.

NASA/CR—2015-218820 75
Chapter 5

Conclusion

A simple analytical method of calculating the discharge coefficient, CD, for undistorted, axisymmetric flows through a MFP is developed from first principles and

verified against experimental results. The control volume method is in excellent

agreement with the test data in the typical operating range (with a maximum error

of 0.54%) of the MFP. The method allows for rapid the geometric variations of typical

mass flow plugs operating under a range of flow rates, pressures and temperatures.

It also accurately couples continuity, momentum, energy, an equation of state, and

wall shear. The effects of boundary layer growth and the reduction in cross-sectional

flow area (characterized by the boundary layer displacement thickness) are calculated

using an integral method. The control volume method is of comparable accuracy to

the CFD when compared to the experimental results. The maximum error in the

CFD is 1.35%. The main advantage of the control volume method is the rapid setup

and computational time.

The CFD results are not capable of predicting the pressure decay through the

MFP, with a maximum error of 90%. The CFD models are run on the supercomputer

cluster at NASA Ames using parallel processing. The control volume requires minimal

geometric modeling since no grid is required. Computational time is also reduced by

NASA/CR—2015-218820 76
CHAPTER 5. CONCLUSION

a factor of 100 for each plug position. The code may be used by experimentalists to

accurately calibrate mass flow plugs and other geometries under choked conditions.

The control volume method developed is more reliable and easier to use than CFD.

It is recommended as a tool to use before developing calibration experiments.

The experimental efforts use a mass flow plug previously employed by Boeing to

determine mass flow rate through a low sonic boom inlet. The rig consists of a small

diffuser, a straight length segment with a contoured exit for the flow to settle and a

plug. The flow through the MFP is measured with an ASME nozzle calibrated by

the metrology lab at NASA Glenn. The ASME nozzle calibration was inconsistent

at low Reynolds numbers which can be attributed to the larger error in the control

volume method outside the operating range.

Total pressure distortion was created at the AIP to quantify its effects on the

mass flow rate. Distortion was produced by rapidly prototyped meshes 3 inches

upstream the AIP. The discharge coefficient for distorted conditions was as much

as 30% lower compared to the baseline calibration. It is shown that deviation in

discharge coefficient depends on both the KA2 and DIST distortion descriptors.

The MFP is sensitive to distortion and its effects on the mass flow rate cannot

be accurately determined by correcting the flow by the MFP total pressure. The

MFP is a flawed design since the straight length segment is too short and does not

let the flow to settle. It is only 3 plug diameters in length, while most MFPs are

at least 6. A larger straight length segment is recommended in further testing of

this MFP. The MFP station rakes are also too coarse and not able to capture all the

total pressure variations across the face. In further distortion studies a rake with at

least the same amount of pitot tubes as the AIP rake is recommended. The CFD

distortion study only showed the general trend in mass flow rate but did not provide

any useful insight to classify the distortion. The study uses only total pressure as

the upstream boundary condition, which is not enough to define the flow. A better

NASA/CR—2015-218820 77
CHAPTER 5. CONCLUSION

simulation requires knowledge of the turbulence structure, and a specific distortion

pattern over a range of plug positions.

NASA/CR—2015-218820 78
Appendix A

Method of Solution

A.0.1 Single Varibale Newton-Raphson Method

The first step of the single variable Newton-Raphson iteration method is to write the

governing function in the form left-hand-side = 0. Equation A.41 shows this step,

where x is the variable to solve by iteratively driving the function f to zero.

f(x) = 0 (A.1)

An initial guess fox x is required to start the iteration. The function value is

checked for convergence (to zero) to within a small numerical tolerance range . This

is represented in Equation A.4A. If Equation A.42 is satisfied by the initial value of

x, the solution is converged

|f(x)| <  (A.2)

If the solution is not converged, an iteration scheme is required to determine a

new value of x. The iteration scheme is Equation A.43, where ∆x is the correction

NASA/CR—2015-218820 79
APPENDIX A. METHOD OF SOLUTION

to be applied to the initial guess of x.

∂f

∂x∆x ≈ f (A.3)

The partial derivative for the function can be approximated as a finite difference

by Equations A.44 and A.45.

δ = x (A.4)

∂f

∂x

≈

f(x + δ) − f(x − δ)

(x + δ) − (x − δ)

(A.5)

The iteration correction step size is calculated by Equation A.46. The iteration

variable is stepped to a new value by Equation A.47. The process is repeated from

Equation A.42 until convergence is reached.

∆x = f/∂f

∂x (A.6)

x = x − ∆x (A.7)

A.0.2 Multi-Variable Newton-Raphson Method

Similar to the single-variable Newton-Raphson iteration method, the first step in

the multi-variable Newton-Raphson iteration method is to write all the governing

equations as left-hand-side equal to zero.

NASA/CR—2015-218820 80
APPENDIX A. METHOD OF SOLUTION

f1 = (x1, x2, x3, . . . , xn)

f2 = (x1, x2, x3, . . . , xn)

. . .

fn = (x1, x2, x3, . . . , xn) (A.8)

The functions fi are evaluated at the initial guesses for the vector of unknowns, xj.

The results are then compared against the specified convergence criteria,. If Equation

A.9 is satisfied, the solutions are converged and no further iteration is required.

|fi(xj )| < , i = 1, 2, 3, . . . , n (A.9)

If the convergence is not met by every function, then an iteration scheme is required to determine the step sizes for each of the unknowns. The iteration scheme is

specified by Equation A.10 in matrix notation.



∂fi

∂xj

  ∆x

≈



f



(A.10)

Equation A.10 is expressed as Equation A.11 and A.12. In this equation, (h

∂fi

∂xj

i

)

is the partial derivative matrix [A], and it is used to solve for the column vector [∆x].

The partial derivative matrix is evaluated numerically by a procedure similar given

in Equations A.5 and A.6. aij refers to elements of [A].

[A][∆x] = [f] (A.11)

aij =

∂fi

∂xj

, i = 1, 2, 3, . . . , n, j = 1, 2, 3, . . . , n (A.12)

NASA/CR—2015-218820 81
APPENDIX A. METHOD OF SOLUTION

The linear system created in Equation A.11 is solved by Lower-Upper decomposition. This procedure is given by Equations A.13 through A.16. Once the corrections

∆xj are determined, the solution is stepper toward convergence by Equation A.17.

The procedure is repeated from Equation A.9 until convergence is reached.

[A] = [L][U] (A.13)

[L]



[U][∆x]



= [f] (A.14)

[L][y] = [f] (A.15)

[U][∆x] = [y] (A.16)

xj = xj − ∆xj, j = 1, 2, 3, . . . , n (A.17)

A.0.3 LU Decomposition

The LU decomposition method is used to solve diagonally-dominant matrix systems.

It assumes that [A] can be decomposed into lower and upper triangular matrices, [L]

and [U] respectively, as shown in Equations A.13 and A.17.



















a11 a12 · · · a1n

a21 a22 · · · a2n

.

.

.

.

.

.

.

.

.

.

.

.

an1 an2 · · · ann



















=



















l11 0 · · · 0

l21 l22 · · · 0

.

.

.

.

.

.

.

.

.

.

.

.

ln1 ln2 · · · lnn





































1 u12 · · · u1n

0 1 · · · u2n

.

.

.

.

.

.

.

.

.

.

.

.

0 0 · · · 1



















(A.18)

NASA/CR—2015-218820 82
APPENDIX A. METHOD OF SOLUTION

The details of decomposing [L] and [U] and calculating the specific elements are

given in Equations A.19 through A.22.

lij = aij −

X

j−1

k=1

likukj , j ≤ i, i = 1, 2, 3, . . . , n (A.19)

uij =

aij −

i

P−1

k=1

likukj

lii

, i ≤ j, j = 2, 3, 4, . . . , n (A.20)

j = 1, li1 = ai1 (A.21)

i = 1, u1j =

a1j

l11

=

a1j

a11

(A.22)

The vector [y] in Equation A.15 is solved by forward substitution using the triangular system. The forward substitution is defined by:

y1 =

f1

l11

(A.23)

yi =

fi −

i

P−1

k=1

likyk

lii

, i = 2, 3, 4, . . . , n (A.24)

Finally, the correction vector [∆x] is solved by backwards substitution of the

triangular system using the intermediate vector [y]. The backwards substitution is

defined by:

∆xn = yn (A.25)

∆xj = yj −

X

j+1

k=n

ujk∆xk, j = n − 1, n − 2, n − 3, . . . , 1 (A.26)

NASA/CR—2015-218820 83
APPENDIX A. METHOD OF SOLUTION

A.0.4 Fourth Order Runge-Kutta

The 4th order Runge-Kutta integration technique is used to solved a set of N coupled

first-order ordinary differential equations for the functions yi, having the general form

dyi(x)

dx = fi(x, y1, . . . , yn) i = 1, . . . , N (A.27)

where the functions fi on the right-hand-side are known.

The Runge-Kutta method propagates a solution over an interval by combining

the information from several Euler-style steps (each involving one evaluation of the

right-hand functions), and then using the information obtained to match a Taylor

series expansion up to some higher order.

Initial conditions for the functions must be specified in order for the method to

start marching forward. The 4th order Runge-Kutta method requires four evaluations

of the right-hand side per step h. The derivative is evaluated at the initial point, twice

at trial midpoints, and once at a trial endpoint. The calculations are presented in

Equations A.27 through A.30.

k1 = f(xn, yn) (A.28)

k2 = f



xn +

h

2

, yn +

k1

2



(A.29)

k3 = f



xn +

h

2

, yn +

k2

2



(A.30)

k4 = f(xn + h, yn + k3) (A.31)

NASA/CR—2015-218820 84
APPENDIX A. METHOD OF SOLUTION

From these derivatives the final function value is calculated.

yn+1 = yn +

h

6

(k1 + 2k2 + 2k3 + k4) (A.32)

The value of x is marched forward by the small step h.

xn+1 = xn + h (A.33)

The solution keeps marching forward until the desired value of y is reached. The

Runge-Kutta method treats every step in a sequence of steps in an identical manner.

Prior behavior of a solution is ot used in its propagation. Mathematically it agrees

with the behavior of ordinary differential equations; any point along the trajectory of

an ODE can serve as an initial point.

A.0.5 Bicubic Spline Interpolation

The goal of the cubic spline interpolation is to get an interpolation formula that is

smooth in the first derivative, and continuous in the second derivative, both within

an interval and at its boundaries. Equation A.34 states the main relation that accomplishes this goal (for j = 2, . . . , n − 1).

xj − xj−1

6

y

00

j−1 +

xj+1 − xj−1

3

y

00

j +

xj+1 − xj

6

y

00

j+1 =

yj+1 − yj

xj+1 − xj

−

yj − yj−1

xj − xj−1

(A.34)

Since Equation A.34 provides n − 2 linear equations in the N unknowns y

00

i

, i =

1, . . . , n, the boundary conditions at x1 and xn must be specified. The natural cubic

spline is specified setting both y

00

1

and y

00

n

equal to zero. Equation A.34 and the

boundary conditions are is not only a liner equation, but also tridiagonal. Each y

00

j

is coupled only to its nearest neighbors at j ± 1. Therefore, the system of equations

NASA/CR—2015-218820 85
APPENDIX A. METHOD OF SOLUTION

can be solved by O(n) operations by a tridiagonal algorithm.

Equations A.35 through A.39 the decomposition of the tridiagonal system and

Equation 2.80 is the backward substitution. The decomposition and substitution

implemented in this section is similar to the LU decomposition.

σ =

xi − xi−1

xi+1 − xi−1

(A.35)

p = σy00

i−1 + 2 (A.36)

y

00

i =

σ − 1

p

(A.37)

ui = 6

yi+1−yi

xi+1−xi

−

yi−yi−1

xi−xi−1

xi+1 − xi−1

− σ

ui−1

p

(A.38)

y

00

k = y

00

k

y

00

k+1 + uk (A.39)

i = 2, ..., n − 1 k = n − 1, n − 2, ..., 1

ui = un = 0

Although the process above describes interpolation in one-dimension, it is possible

to use it as a bicubic spline on an m by n tabulated function. In this case, m

one-dimensional splines are performed across the rows of the table, followed by one

additional one-dimensional spline down the newly created column.

NASA/CR—2015-218820 86
Appendix B

Uncertainty Analysis

If z is a function of x and y (z = f(x, y)), then the error of z depends on the error in

x (∆x) and y (∆y). The error of z is

∆z

2 =



∂f

∂x2

∆x

2 +



∂f

∂y 2

∆y

2

(B.1)

This equation is used to find the experimental error in the discharge coefficient.

The error in the measured flow rate is

∆  ̇m2

actual =



BCDA

√

TT

2

∆P

2

T +



BCDPT

√

TT

2

∆A

2

ASME (B.2)

+



BPTA

√

TT

2

∆C

2

D +

 

BCDAPT

2T

3/2

T

!2

∆T

2

T

where B equals p γ

R



γ+1

2

−

γ+1

2(γ−1)

.

The error in the ASME nozzle throat area is

∆A

2

ASME =



πDASME

2

2

∆D

2

ASME (B.3)

The error in the discharge coefficient depends on the error in the actual flow rate

and the error in the MFP throat area.

NASA/CR—2015-218820 87
APPENDIX B. UNCERTAINTY ANALYSIS

∆C

2

D =

 √

TT

PTAthroatB

2

∆  ̇m2

actual +



−m ̇

2

actual

√

TT

PTA2

throatB

2

∆A

2

throat (B.4)

+



−m ̇

2

actual

√

TT

P

2

TAthroatB

2

∆P

2

T +



m ̇

2

actual

2PTAthroatB

√

TT

2

∆T

2

T

The error in the throat area is

∆A

2

throat = (4.781xplug + 1.804)2∆x

2

plug (B.5)

Finally, the error in the weighted flow is given by

∆w

2 =

 

14.696

PT

r

TT

518.7

!2

∆  ̇m2 +

 

m ̇

14.696

P

2

T

r

TT

518.7

!2

∆P

2

T

(B.6)

+



m ̇ 14.696

√

2PT TT 518.7

2

∆T

2

T

NASA/CR—2015-218820 88
Appendix C

Engineering Drawings

NASA/CR—2015-218820 89
APPENDIX C. ENGINEERING DRAWINGS

Figure C.1: MFP calibration rig assembly.

NASA/CR—2015-218820 90
APPENDIX C. ENGINEERING DRAWINGS

Figure C.2: MFP support assembly- part 1.

NASA/CR—2015-218820 91
APPENDIX C. ENGINEERING DRAWINGS

Figure C.3: MFP support assembly- part 2.

NASA/CR—2015-218820 92
APPENDIX C. ENGINEERING DRAWINGS

Figure C.4: MFP support assembly- part 3.

NASA/CR—2015-218820 93
APPENDIX C. ENGINEERING DRAWINGS

Figure C.5: Settling chamber- part 1.

NASA/CR—2015-218820 94
APPENDIX C. ENGINEERING DRAWINGS

Figure C.6: Settling chamber- part 2.

NASA/CR—2015-218820 95
APPENDIX C. ENGINEERING DRAWINGS

Figure C.7: Settling chamber- part 3.

NASA/CR—2015-218820 96
APPENDIX C. ENGINEERING DRAWINGS

Figure C.8: Connecting flange.

NASA/CR—2015-218820 97
APPENDIX C. ENGINEERING DRAWINGS

Figure C.9: ASME nozzle- part 1.

NASA/CR—2015-218820 98
APPENDIX C. ENGINEERING DRAWINGS

Figure C.10: ASME nozzle- part 2.

NASA/CR—2015-218820 99
APPENDIX C. ENGINEERING DRAWINGS

Figure C.11: ASME nozzle- part 3.

NASA/CR—2015-218820 100
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NASA/CR—2015-218820 101
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NASA/CR—2015-218820 102