Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles

doi:10.4236/msa.2011.210200

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Materials Sciences and Applicatio ns, 2011, 2, 1485-1490

doi:10.4236/msa.2011.210200 Published Online October 2011 (http://www.SciRP.org/journal/msa)

1485

Analysis of Material Behavior for Friction in a

Nozzle for Turbomachinery and High

Speed Vehicles

S. M. Prabhu1*, Abbas Mohadeen2

1NIMS of Engineering & Technology, Jaipur, India; 2Mamallan Institute of Technology, Sriperumbudur, India.

Email: *[email protected], [email protected]

Received January 17th, 2011; revised April 6th, 2011; accepted July 20th, 2011.

ABSTRACT

Shock-induced separation of turbulent boundary layers represents a long-studied problem in compressible flow, bear-

ing, for example, on applications in high speed aerodynamics, rocketry, wind tunnel design, and turbomachinery. Ex-

perimental investigations have generally sought to expose essential physics using geometrically simple configurations.

Keywords: Adiabatic Proces, Friction Efects, Nozzle Modeling, Materialbehavior

1. Introduction

Shock-induced separation of turbulent boundary layers

represents a long-studied problem in compressible flow,

bearing, for example, on applications in high speed

aerodynamics, rocketry, wind tunnel design, and turbo

machinery. Experimental investigations have generally

sought to expose essential physics using geometrically

simple configurations, e.g., supersonic flow over com-

pression ramps [1-4], curved surfaces [2], backward and

forward facing steps [2], simplified wing shapes [5], and

various blunt objects [4,6,7]. While a variety of compu-

tational and analytical methods have also been developed

for treating the problem, the methods are typically appli-

cable to specific compressible flow regimes, i.e., tran-

sonic, supersonic or hypersonic flow, and moreover, due

to the intrinsic unsteadiness of the separation process,

require problem-specific tuning.

The rocket nozzle can surely be described as the

epitome of elegant simplicity. The primary function of a

nozzle is to channel and accelerate the combustion prod-

ucts produced by the burning propellant in such as way

as to maximize the velocity of the exhaust at the exit, to

supersonic velocity. The familiar rocket nozzle, also

known as a convergent-divergent, or de Laval nozzle,

accomplishes this remarkable feat by simple geometry.

In other words, it does this by varying the cross-sectional

area (or diameter) in an exacting form. The analysis of a

rocket nozzle involves the concept of “steady, one-di-

mensional compressible fluid flow of an ideal gas”.

Briefly, this means that:

1) The flow of the fluid (exhaust gases + condensed

particles) is constant and does not change over time dur-

ing the burn.

2) One-dimensional flow means that the direction of

the flow is along a straight line. For a nozzle, the flow is

assumed to be along the axis of symmetry.

3) The flow is compressible. The concept of com-

pressible fluid flow is usually employed for gases mov-

ing at high (usually supersonic) velocity, unlike the con-

cept of incompressible flow, which is used for liquids

and gases moving at a speed well below sonic velocity. A

compressible fluid exhibits significant changes in density,

an incompressible fluid does not.

4) The concept of an ideal gas is a simplifying assump-

tion, one that allows use of a direct relationship between

pressure, density and temperature, which are properties

that are particularly important in analyzing flow through a

nozzle.

Fluid properties, such as velocity, density, pressure

and temperature, in compressible fluid flow, are affected

1) Cross-sectional area change.

2) Friction.

3) Heat loss to the surroundings.

Experimental Setup

Figure 1 shows the Experimental setup of Nozzle under

Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles

1486

adabatic flow conditions. The heat flux distribution is

determined by thermocouple connected on the work

piece at definite distances. Cu-Ni Thermocouples were

attached on the surface of the work piece. The flame was

generated by acetylene and oxygen gasses as in gas

welding process. The flame was kept at distances from 5

cm to 20 cm and the heat flux distribution studied. Dur-

ing a typical experimental run the powers were varied to

achieve different base late temperature and hence Ray-

leigh numbers. Due to temperature constraints the pa-

rameters of the heat input were restricted to maximum

base plate temperature of 250˚C.

2. Model Proposal

This paper investigates time-average, shock-induced tur-

bulent boundary layer separation in over-expanded rocket

nozzles. Although focused on this particular problem,

much of the development applies to the same broad fam-

ily of shock-separated flows encompassed by the free

interaction model. The objectives are to first present and

examine an alternative to the free interaction model, with

a view toward obtaining a fuller understanding of the free

interaction process. Simple scale analyses of transverse

momentum transport across the separating boundary

layer are presented and used to derive criteria for esti-

mating the approximate time-average separation pressure

ratio, Pi/Pp, as a function of the in viscid separation Mach

Number, Mi = M(xi), where Pi is the time-average wall

pressure at the point of incipient separation, xi , and Pp is

the peak wall pressure at the downstream limit of the

shock interaction zone, xp; see Figure 1. In the case of

rocket nozzle flows, where separation-induced side

loading constitutes an intrinsic feature of low altitude

flight [15,16], knowledge of the separation criterion Pi





 (1)

is crucial since it allows determination of the corre-

sponding separation line location. The time average

pressure gradient over the shock interaction zone (xi-

x-xp), given approximately by







 (2)

in reality reflects the intermittent, random motion of the

shock between xi and xp [3]. As the shock-compression

wave system oscillates randomly above (and partially

within) the boundary layer, the associated pressure jump

across the system is transmitted across the boundary

layer on a time scale



(a)

(b)

Figure 1. (a) Experimental setup of Nozzle under adabatic

flow conditions; (b) drag coefficient for sphere and cylinder

in hypersonic flow.

tion point essentially tracks the random position of the

shock-compression wave system, where the position of

the separation point is described by a Gaussian distribu-

tion over the length of the interaction zone [3].

Scale Analysis I

Considering the vertical momentum balance immediately

downstream of the separation point xs, it is recognized

that the boundary layer lifts off of the wall due to a ver-

tical gradient in pressure [2,30]. Thus, the vertical advec-

tion of vertical momentum must be of the order of the

vertical pressure gradient:

vvp









 (3)

or in approximate form,

vvy







 (4)

where the density ρ2 of the boundary layer near xs is ap-

proximated as the free stream density downstream of the

shock,vs is the characteristic vertical velocity component

within the boundary layer near xs, and δs ≈ δi is the char-

acteristic boundary layer thickness near xsPp − P2, is es-

timated as the difference between the peak wall pressure,

kRT



, where δi and Ti are

the characteristic boundary layer thickness and tempera-

ture in the vicinity of xi. Under typical experimental condi-

tions, Ts is much shorter than the slow time scale, f = 1s

(where Ts ≈ 1 to 10 µs); thus, the instantaneous separa-

Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles1487

Pp, in the vicinity of xp and the free stream pressure im-

mediately downstream of the oblique shock. The density

estimate in (4) recognizes that the boundary layer has

passed through the compression wave system at the foot

of the oblique shock.

Eliminating δs from (4) and solving for vs yields



€

vv1



 (5)

From Figure 2, we note that at the separation point, vs

is related to the characteristic horizontal velocity com-

ponent, us by

tan ,



 (6)

where θ is the characteristic angle of deflection between

the separating boundary layer and the nozzle wall. The

magnitude of us is estimated by again noting that the

boundary layer flow has passed through the compression

wave system at the foot of the oblique shock and that, as

indicated in Figure 2, the time average turbulent bound-

ary layer velocity profile is nearly flat. Thus, us is on the

order of the x-component (U2) of the inviscid flow veloc-

ity (v2) immediately downstream of the oblique shock:

2cos





 (7)

Using the ideal gas relation, ρ2 = kP 2/(kRT2), and in-

serting (5) in (6) we then obtain

0.5

tan p

















(8)

where M2x = M2 cos θ, and M2 is the in viscid flow Mach

Number immediately downstream of the oblique shock.

Rewriting Pp/P2 as (Pp/P1)(P1/P2) and solving (8) for

Pp/P1 finally yields

Figure 2. Plot of nose length vs friction adiabatic coefficient

of the nozzle.

1sin







(9)

Identifying P1/Pp as the critical wall pressure ratio at

which separation occurs, i.e., P1/Pp ≈ Pi/Pp, noting that

M2 is given by the oblique shock relation









222

1sin 2

2sin 21

kM k

MkM k







 (10)

and recognizing that Mi ≈ M1, where M1 is the free stream

Mach Number immediately upstream of the oblique

shock, it is seen that (9) provides an explicit, physically-

based relationship between Pi/Pp and the pressure ratio,

P1/P2, across the oblique shock. The next scale analysis

refines the estimates for stream wise inertia and cross-

layer pressure gradient, and leads to a near-identity be-

tween Pi/Pp and P1/P2.

2.1. Optimization of Contour Nozzle Design

Including Viscous Effects

Here we present a genetic algorithm for design of Mach

12 contoured Nozzle. The objective is to produce a uni-

form velocity (Mach Number) over the entrance region

in the nozzle exit plane and to limit the flow angularity at

the exit plane, over the same region within a given limit.

At hypersonic wind tunnel nozzles with Mach Numbers

greater than 8 are dominated with strong viscous effects,

the nozzle contour generated by conventional method of

characteristics does not meet the design requirements

when boundary layer corrections are made. In the present

work parallel Aerodynamic simulator code (PARAS) or

FLUENT. The former uses the surface oriented mesh

system to simulate the flow inside the axisymmetric noz-

zle. The code solves Navier stokes equations using a Fi-

nite volume approach and is convenient and fast. To rep-

resent the nozzle contour generated by conventional

method of characteristics. CFD solutions are used to

evaluate the objective function in each function evalua-

tion of the Genetic Algorithm (GA) process. To represent

the nozzle contour in terms of certain parameter vector

P(p1, p2, ···, pn) the nozzle contour is divided into 5 seg-

ments and is represented as follows.:

For 0 < x  x1 r = a0 + a1x + a2x2 + a3x3

x1 < x  x2 r = a0 + a1x + a2x2 + a3x3

x2 < x  x3 r = a0 + a1x + a2x2 + a3x3

x3 < x  x4 r = a0 + a1x + a2x2 + a3x3

x4 < x  x5 r = a0 + a1x + a2x2 + a3x3

The above five cubic spheres result in 20 coefficients

(a0, a1, ···, a20) assuming that the locations x1, x2, ···, x5

are known. The slopes at the nozzle throat and the exit

plane are assumed to be zero. The continuity of radius,

slope and curvature at the four interfaces give 4 condi-

Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles

1488

tions. With these 20 conditions, the above set of linear

simultaneous equations is solved using the Gauss Siedel

method. After getting the coefficients a0, ···, a19 the noz-

zle contour can be determined. The length of the nozzle

is also kept as a floating parmeter. A constant inlet Mach

Number of unity has been taken and the tunnel operating

conditions have been taken as p0 = 68 KSC, T0 = 1500 K

and ratio of specific heats



= 1.31. Besides the radii rth,

r1, r2, r3, and r4 the length of the nozzle L (=x5) is also

considered as a parameter. The nozzle exit radius is 0.5

m which is the test section radius is 0.5 m which is the

test section radius, and is fixed. The nominal values of

these parameters are arrived at using the MOC contour

with the boundary layer correction.

The objective function consists of 2 parts. First part

takes into account the deviation of actual Mach Number

Mact from the target Mach Number Mtar (=12)and the sec-

ond part of the deviation of flow angularity



act from the

target flow angularity



tar (0.2 deg). These are evaluated

at the core of the nozzle exit. Thus we have (Mtar―Mach

at target)





acttaract tar

Obj1 M

PN MM







 









where P is the parameter vector and N is number of grid

cells in the core of the nozzle exit. Values of



M and





are taken as 0.7 and 0.28 respectively, which are the

weighting factors.

Figure 7 shows the Objective function as defined

above with number of generations. It is seen that the ob-

jective function falls by 30% in about 68 generations.

2.2. Modeling of Hypersonic Flows

The Navier Stokes equations for the hypersonic flow are

given by the continuity and momentum equations.

Continuity Equation





 (11)

vvgk

uv T



 



 x

(12)

where volumetric heat sources (x, y, z) represents the

contribution of frictional heating. The parameters



Cp &

ke may depend on y & z but remain independent of x.

More importantly the contributing of axial conduction

deferred to the subsequent is neglected, hence Equation

(4) reduces



22 0

uT vTTTuv

xx KCp







 

 



 

 (13)

When these are non dimensionalized g the following

definitions of variables

** **

inf

*2*

, , , ,

=, =

xl yyl zzl uuV

PP v



 



we get

** 0u



 (14)



**

(15)



 (16)

For flow tangent to body V·n = 0.

Let nx, ny, nz be the vector. Using the final pressure ra-

tio equations can be derived for the nozzle as below.

Consider the hyperonic flow over a given body in the

limit of large M. The flow is again governed by Equa-

tions (4)-(6) we get the Bcondns as



2sinp









(17)









 (18)







212sin1u









(19)



2sin 1v



 (20)

From this consideration we can seethe Mach Number

is independent and this follows from the governing Equa-

tions of motion with the appropriate BCs. Written in

limit of high Mach Number. Hence when the free stream

Mach Number is very high the dimensionless variables

become independent on Mach Number, this trend applies

to any quantities derived from these dimensionless vari-

ables. The drag coefficient follows a plot as in Figure 1

below.

Examining the governing flow Equation upon which

hyperbolic similarity is based (Equations of continuity &

motion) in dimensionless form)The similarity principle

holds for both irrotational and rotational flow as shown

in Figure 2 where the two curves for irrotational & rota-

tional flow overlap each other. The surface pressure dis-

tribution is shown in Figure below.

The sphere drag achieve Mach no independence at

lower Ma. For blunt cone end ogive cylinders the veloc-

ity distributions is given below.

The maximum location of the shock is btaies as shown

in Figures 4 and 5.

The main point in this discussion however, to find the

farthest shock location downstream. Figure 5 shows the

possible as function of retreat of the location of the shock

wave from the maximum location. When the entrance

Mach Number is infinity, P2, if the shock location is at

the Maximum length, than shock at M2 < 1 results in

Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles1489

4fL/D = 0.3 and possible.

The proposed procedure is based on Figure 5.

1) Calculated the extra 4fL/D and subtract the actual

extra 4fL/D assuming shock at the left side (at the max

length).

2) Calculated the extra 4fL/D and subtract the actual

extra 4fL/D D assuming shock at the right side (at the

entrance).

3) According to the positive or negative utilizes your

root finding procedure.

From numerical point of view, the Mach Number

equal infinity when left side assume result in infinity

length of possible extra (the whole flow in the tube is

subsonic). To overcome this numerical problem it is

suggested to start the calculation from distance from the

right hand side.

Let us denote

4fL/D > 4fLmax /D~0.34

Note that 4fL/D < 4fL/D ma x is smaller than P2. The re-

quirement that has to satisfied is that denote as difference

between the maximum possible of length in which the

flow supersonic achieved and the actual length in which

the flow is supersonic as in Figure 8. The retreating length

is expressed as subsonic but as the Figure 8 shows the

entrance Mach Number, M1 is reduced after the maximum

length is exceeded. From numerical point of view, the

Mach Number equal infinity when left side assume result

in infinity length of possible extra (the whole flow in the

tube is subsonic). To overcome this numerical problem it

is suggested to start the calculation from distance from the

right hand side.

3. Results and Discussion

Figure 2 shows the plot of Nose length vs friction adia-

batic coefficient of the nozzle Figure 3 gives the Pres-

sure distributions for ogive cylinders: illustration of hy-

personic similarity a) K = 0.5 b) K = 1.0. Figure 4 shows

the Surface pressure distribution at x/Rn = 8



= 20*,



1.67, Re = 86000,



c = 15. Figure 5 Pressure distribution

over blunt nosed cone, compared with pointed cone. Fig-

ure 6 shows the Fanno flow characteristics.

Figure 3. Pressure distributions for ogive cylinders: illus-

tration of hypersonic similarity (a) K = 0.5 (b) K = 1.0.



Figure 4. Surfac e pressure distribution at x/Rn = 8,



= 20*,



= 1.67, Re = 86000,



c = 15.



Figure 5. Pressure distribution over blunt nosed cone, com-

pared with pointed cone.

Figure 7 depicts the Mach Number variation with fric-

tion headloss. Figure 8 gives the depiction of pressure

drop variation with friction headloss. At hypersonic wind

tunnel nozzles with Mach Numbers greater than 8 are

dominated with strong viscous effects, the nozzle contour

generated by conventional method of characteristics does

not meet the design requirements when boundary layer

corrections are made. In the present work parallel aero-

dynamic simulator code (PARAS) or FLUENT. The re-

quirement that has to satisfied is that denote as difference

between the maximum possible of length in which the

flow supersonic achieved and the actual length in which

the flow is supersonic as in Figure 8. From numerical

point of view, the Mach Number equal infinity when left

side assume result in infinity length of possible extra (the

whole flow in the tube is subsonic).

4. Conclusions and Suggestions

Time-average, shock-induced, turbulent boundary layer

separation has been investigated using a combination of

heuristics, simple analytical models, and experiments,

with a focus on separation in over expanded rocket noz-

zles. Two simple scaling analyses are presented in which

Analysis of Material Behavior for Friction in a Nozzle for Turbomachinery and High Speed Vehicles

1490

Figure 8. Plot of Pressure ratio in the nozzle to aspect ratio

of nozzle.

Figure 6. Fanno flow characteristic s. sure ratio is, to a good approximation, determined by the

oblique shock pressure ratio.

REFERENCES

[1] G. Settles, “Details of a Shock-Separated Turbulent Bound-

ary Layer at a Compression Corner,” AIAA Journal, Vol.

14, No. 12, 1976, pp. 1709-1715. doi:10.2514/3.61513

[2] D. R. Chapman, D. M. Kuehn and H. K. Larson, “Inves-

tigation of Separated Flows in Supersonic and Subsonic

Streams with Emphasis on the Effect of Transition,”

NACA Report 1356, 1958.

[3] M. E. Erengil and D. S. Dolling, “Correlation of Separa-

tion Shock Motion with Pressure Fluctuations in the In-

coming Boundary Layer,” AIAA Journal, Vol. 29, No. 11,

1990, pp. 1868-1877. doi:10.2514/3.10812

[4] D. S. Dolling and L. Brusniak, “Separation Shock Motion

in Fin, Cylinder, and Compression Ramp-Induced Turbu-

lent Interactions,” AIAA Journal, Vol. 27, No. 6, 1988, pp.

734-742. doi:10.2514/3.10173

[5] J. Ackeret, F. Feldman and N. Rott, “Investigations of

Compression Shocks and Boundary Layers in Gases

Moving at High Speed,” NACA TM No. 1113, 1947.

Figure 7. Mach Number variation with friction head loss.

[6] D. S. Dolling and D. R. Smith, “Separation Shock Dynam-

ics in Mach 5 Turbulent Interactions Induced by Cylin-

ders,” AIAA Journal, Vol. 27, No. 12, 1989, pp. 1698-1706.

doi:10.2514/3.10323

separation is viewed as reflecting a balance between

streamwise boundary layer inertia and the cross-layer

pressure gradient. These lead to two theoretical expres-

sions for the time average separation pressure ratio,

stated as a function of the inviscid Mach Number at the

point of incipient separation. In both models, explicit

relationships between the separation pressure ratio (at the

wall) and the classical oblique shock ratio are obtained;

the second model, representing a refinement of the first,

emonstrates for the first time that the separation pres-

[7] R. A. Gramann and D. S. Dolling, “Detection of Turbu-

lent Boundary-Layer Separation Using Fluctuating Wall

Pressure Signals,” AIAA Journal, Vol. 28, No. 6, 1989, pp.

1052-1056. doi:10.2514/3.25164

[8] J. Ostlund, “Flow Processes in Rocket Engine Nozzles

with Focus on Flow-Separation and Side-Loads,” Licen-

tiate Thesis, Royal Inst. of Technology, Stockholm,

TRITA-MEK, Technique Report, 2002, p. 1112.