- :2

NASA Technical Memorandum 102397

STOVL Propulsion System Volume

Dynamics Approximations

Colin K. Drummond

Lewis Research Center

Cleveland, Ohio

Prepared for the

1989 Winter Annual Meeting

sponsored by the American Society of Mechanical Engineers

San Francisco, California, December 10-15, 1989

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STOVL PROPULSION SYSTEM VOLUME DYNAMICS APPROXIMATIONS

Colin K. Drummond

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

ABSTRACT

Two approaches to modeling turbofan engine component volume dynamics are explored

and compared with a view toward application to real-time simulation of short take-off vertical

landing (STOVL) aircraft propulsion systems. The first (and most popular) approach considers

only heat and mass balances; the second approach includes a momentum balance and substitutes

the heat equation with a complete energy balance. Results for a practical test case are presented

and discussed.

INTRODUCTION

An accurate propulsion system simulation for short take-off vertical landing (STOVL)

aircraft is a critical element for research on design methodologies for integrated STOVL aircraft

flight and propulsion control systems. Such a s_mulation must, in principle, mimic steady-state

and transient sub-system performance and be executable in real-time. For modern real-time

system analyses, it is important that the STOVL simulation include the dynamics of component

feeder ducts, engine inter-component volumes, and the tailpipe.

Early engine simulations on analog computers often included intercomponent volume

dynamics modules, but the intent was more often for the purpose of coupling static components

than for an accurate solicitation of gas dynamics effects. As such, the traditional approach draws

only on the continuity equation and a simplified form of the energy equation. Later, simulations

on digital computers relaxed the "complete" form of the initial value problem by dropping

intercomponent approximations and employed iterative schemes to balance the engine. The

latter approach is not appropriate for simulations on modern real-time computers (such as the

Applied Dynamics International AD-100 machine) and is the impetus for the present investigatmn. Problem formulations for real-time STOVL systems must contain accurate volume

dynamics approximations.

Feeder ducts and tailpipes of STOVL engine configurations are likely to exhibit different

dynamic behavior than inter-component volumes and therefore may not be accurately characterized by traditional (analog) volume dynamics representations. There is a need for a volume

dynamics model with the simplicity and computational speed of the traditional approach, but

with the capability to incorporate more detailed heat transfer and fluid-dynamic effects that are

likely features of STOVL systems.
In the present work, two approachesto modeling volume dynamicsare explored and

comparedwith aview toward apphcationto real-time propulsion systemsimulation. The first

andsimplestmethodemploysthe traditional approachin theproblemformulation; in thesecond

approacha momentum oalance is included and the heat equation is replaced with a complete

energy balance. These approaches are applied to a typical ejector feeder pipe for a proposed

ejector-configured STOVL propulsion system. A third method under development is also

mentioned which employs an entropy balance in place of the energy equation.

PROBLEM FORMULATION

Some traditional approaches to the transient volume dynamics problem are represented

by the lumped volume work of Szuch et al.[1], the finite difference analysis of Cheng and

Bowyer[2] and the control volume approach of Thompson[3].

In order to more easily accommodate multiple inlet and exit flows, the present approximations for mass, momentum, and energy balances are discussed within a control-volume

framework. An integral approach allows complicated aero-thermodynamic physics to be more

simply and robustl_¢ be described by a gross relation involving a group of particles rather than,

say, a flow streamline approach.

For the intent of the present simulation, it is reasonable to proceed with a one-dimensional

gas flow assumption. This sets the stage for simplified control volume equations. Consider the

control-volume k shown in Fig. 1. Multiple inlets are permissable at surface i and multiple

discharges atj. To apply the assumptions above, first define control-volume mass flux as

m = -fspu.n_aA (1)

so, for example, the mass influx is

N

m, = + _. P.vnA,, (2)

n-|

The governing balances of mass, momentum, energy, and entropy can now be written in the

form

d fvpdV=rtz_-mj (3)

-_t fvPU_clV = {v,rn,- u/m,,} + {p,A,- p jA /} + F (4)

t,LP (ho- dV=miho,i-rnjho,i (U)

a fv BQ_'s > o (6) psdV-m_s_+m/s/ T -

Further simplification of the system of equations hinges on discussion regarding the choice of

state variables and the treatment of the time-dependent volume integrals.
BASIC HEAT EQUATION ANALYSIS

Two particularly simple differential equations for the basic heat equation (BHE) analysis

result from the application of two assumpnons. The first assumption ts that changes in flow

kinetic energy are negligible; this reduces the first law of thermodynamics to the heat equation.

A very basic analysis follows from the second assumption in which momentum changes are

neglected. Since many mechanical effects that influence the flow are discarded, we expect the

approach is most appropriate for short ducts of nearly uniform cross-section.

For a spatially uniform density approximation the volume integral in eqn.(3) can be

evaluated. If it is also assumed the volume of region k is of fixed size, the governing equation

for the density derivative is

dp) = m,-m.j = m,-mj (7) k Vk _Ax

where an overbar indicates a volume-average quantity.

In the absence of mechanisms to influence momentum loss, state chan_.es arise through

changes in thermal energy, not mechanical energy. For a zero momentum deficit, the pressure

within a given control volume is subsequently uniform (quasi-steady), and the computation for

P is simply

P = pRT (8)

Changes in pressure are therefore a direct result of changes in density and temperature.

When the mechanical power equation (the dot product of the velocity and Navier-Stokes

equation) is subtracted from the general power equation (the first law of thermodynamics, eqn.5)

there results the heat equation; in integral form the adiabatic result is

podV = - pev.ndA- pv.ndA (9)

In the ideal case of a perfectly mixed one-dimensional flow of small axial temperature gradient

(and a uniform pressure throughout), an average enthalpy h can be introduced. This yields

the traditional result for the temperature derivative

ctT) _ ¥- 1 {m,(h _-_)+ RT(m _mi) } (lO) k Rmcv

Equations (7) and (10) constitute the differential equations for the basic heat equation analysis,

and are the typical equations one might see in early analog simulations of engine inter-component

volumes.

MOMENTUM/ENERGY EQUATION ANALYSIS

The momentum/energy equation (MEE) approach to the duct analysis includes a balance

of momentum and replaces the simple heat equation of the BHE approach with the general

energy equation.
Flow momentum changes in ducts are generally attributable to variations in cross-sectional

area or the friction associated with real flows; the governin,_ differential equation is a vector

expression, but with the flux momentum nomenclature of Fig. 1 in mind (see [3] for details), there results the scalar balance

d (pVV)k = Mi- M j + fi (11)

where the scalar form of the momentum is

M = vm + PA

In this work an idealized expression for the internal resistance of the duct shown is given by

- 1

P = PdA _ 2"''_-(P'+Pi_rAj-A'_

where the absence of frictional effects should be noted. Carrying through the differentiation in

time and re-arranging results in the following equation

-6-[) do _ __1 (Mi_Mj+I_)___.._ a o- (12)

k oAAx pc:It

The characteristic velocity _ is a weighted average of the inlet and exit control volume velocities

i

v = _v,+(1-_)vj

where _ is a weighting parameter ranging in value from 0 to 1.

The general power equation includes representations of thermal and kinetic energy,

(13)

where

The time-dependent term can be rearranged to read

a-t P h+ .... +_v--+

2 p k y-1 dt dt 2 cltJk

and from this an expression for the pressure derivative obtained

4
1 dp = {£,-Ej}- vm+ - (14)

V- I -'_ , dt 2 at k

Equations (7), (12), and (14) are the differential equations of interest in the momentum/energy

equation analysis.

REMARK ON ENTROPY ANALYSIS

Traditionally, the entropy equation is employed as a check on the validity of multiple

solutions (subsonic flow, supersonic flow) extracted from the general energy balance. However,

except for the mathematical inconvenience of the inequalit_¢ in eqn.(6), there should be no reason

the entropy equation cannot replace the general form otthe energy equation. An analysis is

in-progress to investigate replacement of the lost-work-energy term so that

--AAxd(pS)dt m_s,+rit/sj = Cl>0- (15)

where C 1 is an empirical constant (that replaces the lost-work term) associated with a specific

duct configuration. When a reference entropy point has been established, the ideal gas relations

permit evaluation of entropy at any other desired state. Here, the proposed benefit is in condensing the otherwise scattered real flow effects (friction, heat transfer) into one constant,

expediting computations for a given duct. It remains to establish whether computations for

entropy at each point is faster than the quadratic solution previously required for the velocity

(from the energy equation).

RESULTS

Duct system response to step-function inputs are considered in this work since these

changes in duct flowrate are coincident with typical STOVL flight scenarios. In the present work,

flow conditions representative of previous NASA Lewis ejector tests provide baseline data for

a realistic test case. We ,therefore consider a step change in flowrate from 8.48 Kg/s to 9.91 Kg/s

that occurs with an incr,.ase in static pressure from 106 ]_Pa to 108 kPa and at a constant inlet

temperature of 400K. A constant area duct of 0.102 m z is used for all calculations; also, all

numerical calculations proceeded with a time step of I ms. Two cases presented are where the

duct volume is first contained within one finite-volume and then partitioned into several finite

elements.

Our first result points out a potential hazard in simplifying the time derivatives of the

control-volume integrals with the "characteristic" field variables (v, 0). Figure 2 illustrates predictions of exit flowrate for ducts of 1.5m and 4.6m in length. Note the absence of momentum

exchange in the BHE approach essentially provides a quasi-steady duct flowrate response. The

MEE predictions reqmre twice as much computer time and seem more reasonable, but the

discharge of the 1.5m duct which lags its 4.6m counterpart reflects the inaccuracy of the assumed

value of the weighting scalar _ = 0.5 for very long ducts; after one time step the incoming fluid

has penetrated only 2% of the 4.6m duct, not the 50% implied in the calculation. When _ = 0.9

the situation corrects to the shorter duct reaching a steady-state before the longer duct does.

In Fig. 3 it appears the pressure response of the BHE approach is flawed by the absence

of momentum exchange; pressure changes must inadequately communicate through the heat

equation. Since Fig.3 is based on _ = 0.5 the trend in pressure response for the MEE approach

is intuitively reversed, but, as mentioned above, corrects when _j= 0.9. The cautionary note from

5
Figs. 2 and 3 is that althou._h the BHE calculations ap.pear stable in the analysis of very long

ducts, this by no means imphes the results are necessarily accurate. If, however, analytic interest

is focused only on accurate representation of flowrate, use of the BHE approach may then be

appropriate.

Figure 4 illustrates flowrate predictions with the MEE approach for a 3m long duct divided

into 5 control-volumes. In this case the reduced length of the individual control volumes partially

compensates for the inaccuracy of results for the assumed _j = 0.5; generally, the "correct" _ to

use is not known apriori. In practice, 5 control volumes require the same multiple of computations

that a single-volume computation does. It therefore remains to balance in propulsion applications

whether or not the need for machine speed solicits pre-processing _ for greater single-volume

duct analysis accuracy.

It appears from the computations that the BHE apprach is not adequate to describe the

essential transient flow features required to accurately s_mulate propulsion system feeder ducts

and tailpipe responses, but will suffice for traditional inter-stage engine component coupling.

The more detailed (and computationally intensive) MEE approach is necessary for simulating

STOVL-specific propulsion system components.

CONCLUSION

The modeling approaches presented provide rapid characterization of ejector/duct field

variable time dependencies. These models are potentially useful for real-time STOVL propulsion simulation development efforts. If, however, the basic assumptions of each approach

are not appreciated in application, the net result is the rapid computation of the wrong answer.

A

E

P

ho

h

KE

L

m

inca

A,!

A/J cu

P

t?

S

t

lJ

I F

X

Ax"

P

NOMENCLATURE

cross-sectional area

energy flux

internal resistance of the duct

total enthalpy, = h + u 2/2

specific enthalpy

kinetic energy

total duct length

mass flowrate

control volume mass

momentum flux

control volume momentum

pressure

gas constant

entropy

time

velocity

volume of control-volume

streamwise coordinate

duct incremental length

density of gas

weighting parameter for average velocity calculation

6
Subscripts

i

J

k

n

Superscript

inflow

outflow

control volume designation

dummy index

averaged quantity

REFERENCES

1. Szuch,J., Krosel,S., and Bruton,W., "Automated Procedure

Computer Simulations of Turbofan Engines," NASA TP-1851, 1982.

for Developing Hybrid

2. Cheng, L. and Bowyer,J., "TUBE-Transient Compressible Flow Code," Fluid Transients

& Acoustics in Industry., ASME, Dec. 1978.

3. Thompson, P., Compressible Fluid Dynamics, McGraw Hill, New York, 1972.

i k

T,P,v

Wall

E

Wall

--.ID,* X

L

T,P,v

--P" n

Fig.1 Control-volume approximati6n.
t_

e_

v

or}

t_

10.22-

8.76

7.29

Fig.2

= 0.5

BHE,Ax = 4.6m

9.91 kg/s "_

"_ ,-'t"- _ --'------- •

BHE ,Ax = l.Srn

Ax 4.6m

l _MEE,Ax = 1.Sin

---- 8.48 kg/s

| i

0 3 6

BHE - Basic Heat Equation

MEE Momentum/Energy Equation

! | ! I ! ..... | • - _l

9 12 15 18 21 24 27 30

Time (ms),

Solution for duct exit flowrate as function of time.

o

t_

ta_

0

116.2"

111.1

105.3

Fig.3

MEE,Ax = 1.5m

//MEE,Ax = 4.6m

__/_nlet Pressure

/./

,Ax = 1.Sm

,Ax = 4.6m

0

i i ! i | • ! | • •

3 6 9 12 15 18 21 24 27 30

Time (ms)

Solution for duct exit pressure as function of time.
= 0.5 Duct Length = 3m

O

oO

10.22

8.76

Influx

I

m

7.29 [ , ° , , , , , , , ,

0 3 6 9 12 15

Time (ms)

Fig.4 Solution for duct flowrate as function of time for three points along the duct.

Report Documentation Page National Aeronaullcs and

Space Admlntstratlon

1. Report No. 2. Government Accession No. 3. Recipienl's Catalog No.

NASA TM- 102397

4. Title and Subtitle 5. Report Date

STOVL Propulsion System Volume Dynamics Approximations

7. Author(s)

Colin K. Drummond

9. Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

i12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

6. Performing Organization Code

8. Performing Organization Report No.

E-5147

10. Work Unit No.

505-62-71

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared for the 1989 Winter Annual Meeting of the American Society of Mechanical Engineers, San Francisco,

California, December 10-15, 1989.

16. Abstract

Two approaches to modeling turbofan engine component volume dynamics are explored and compared with a

view toward application to real-time simulation of short take-off vertical landing (STOVL) aircraft propulsion

systems. The first (and most popular) approach considers only heat and mass balances; the second approach

includes a momentum balance and substitutes the heat equation with a complete energy balance. Results for a

practical test case are presented and discussed.

17. Key Words (Suggested by Author(s))

STOVL

Real-time simulation

Duct flow

Unsteady flow

t8. Distribution Statement

Unclassified- Unlimited

Subject Category 07

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages

Unclassified Unclassified 11

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