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Flow Analysis in a Convergent-Divergent Nozzle Using CFD

1Gutti Rajeswara Rao,

2U.S. Ramakanth,

3A. Lakshman

1Assistant Professor, St. Marry Engineering College, India 2Associate Professor, VITAM Engineering College, India

3M.Tech CAD/CAM SMBS, VIT University, India

ABSTRACT

The effects of Mach number and Nozzle pressure ratios (NPR) on Mass flow rate, Maximum pressure,

and Maximum velocity and on Maximum force are studied using Fluent Analysis. The classical one

dimensional inviscid theory does not reveal the complex flow features in a convergent divergent

nozzle accurately. The code fluent has been used to compute flow using a coupled and axisymmetric

Convergent Divergent nozzle for different nozzle ratios and for different Mach numbers.

Keywords: Nozzle Pressure Ratio (NPR), Mach number, Convergent-Divergent Nozzle, CFD.

1. INTRODUCTION

In the present days there is a huge development in Aerospace Engineering for defense and civil

prospects. Extensive research is being carried out in these fields. This project tries to establish the

methods of developing the rocket nozzle of a solid propellant rocket. The test is virtualized at

different Mach numbers, where the flow conditions are derived.

The virtualization is one of the major developments in the field of research, which revolutionized

Aerospace engineering, along with all other branches. The computational techniques are used widely

for getting better results, close to experimental techniques.

The flow through a converging-diverging nozzle is one of the benchmark problems used for modeling

the compressible flow through computational fluid dynamics. Occurrence of shock in the flow field

displays one of the most prominent effects of compressibility over fluid flow. Accurate shock

predication is a challenge to the CFD fraternity. In order to resolve the high pressure gradients we

need to use some special numerical schemes along with fine grid. In some cases, local grid adaption

can be helpful. The one-dimensional inviscid isentropic flow in a convergent-divergent nozzle is a

classical text book problem ,which has different flow regimes depending upon the nozzle pressure

ratio(NPR).The inviscid theory predicts a simple shock structure consisting of a normal shock

followed by a smooth recovery to exit pressure in the divergence part of a chocked nozzle for the

nozzle pressure ratios corresponding to the over-expanded flow regimes. But ,in reality, multi

dimensionality and viscous effects like wall boundary layer and flow separation drastically alter the

flow in a CD nozzle. The over-expanded flow regime in CD nozzles of different shapes and sizes has

been a subject matter of numerous investigations because of their wide range of applications. One of

the more recent investigations in an experimental study of flow in rectangular over expanded

supersonic nozzles exploring the complexity nature of such flows. The prediction of such flows also

presents a great challenge to any CFD code.

2. LITERATURE

CONVERGENT-DIVERGENT nozzle is designed for attaining speeds that are greater than speed of sound.

the design of this nozzle came from the area-velocity relation (dA/dV)=-(A/V)(1-M^2) M is the Mach

number ( which means ratio of local speed of flow to the local speed of sound) A is area and V is velocity.

International Journal of Research in Mechanical Engineering

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The following information can be derived from the area-velocity relation -

1. For incompressible flow limit, i.e. for M tends to zero, AV = constant. This is the famous volume

conservation equation or continuity equation for incompressible flow.

2. For M < 1, a decrease in area results in increase of velocity and vice vera. Therefore, the velocity

increases in a convergent duct and decreases in a Divergent duct. This result for compressible subsonic

flows is the same as that for incompressible flow.

3. For M > 1, an increase in area results in increase of velocity and vice versa, i.e. the velocity increases

in a divergent duct and decreases in a convergent duct. This is directly opposite to the behavior of

subsonic flow in divergent and convergent ducts.

4. For M = 1, dA/A = 0, which implies that the location where the Mach number is unity, the area of

the passage is either minimum or maximum. We can easily show that the minimum in area is the only

physically realistic solution.

One important point is that to attain supersonic speeds we have to maintain favorable pressure ratios

across the nozzle. One example is : to attain just sonic speeds at the throat, pressure ratio to e

maintained is (Pthroat / P inlet)=0.528.

Table.1

Regime Subsonic Transonic Sonic Supersonic Hypersonic High-hypersonic

Mach <1.0 0.8-1.2 1.0 1.0-5.0 5.0-10.0 >10.0

From table.1 at transonic speeds, the flow field around the object includes both sub- and supersonic parts.

The transonic period begins when first zones of M>1 flow appear around the object. In case of an airfoil

(such as an aircraft's wing), this typically happens above the wing. Supersonic flow can decelerate back to

subsonic only in a normal shock; this typically happens before the trailing edge. (Fig.a)

As the speed increases, the zone of M>1 flow increases towards both leading and trailing edges. As M=1 is

reached and passed, the normal shock reaches the trailing edge and becomes a weak oblique shock: the

flow decelerates over the shock, but remains supersonic. A normal shock is created ahead of the object,

and the only subsonic zone in the flow field is a small area around the object's leading edge. (Fig.b)

Figure (a) Figure (b)

The governing continuity, momentum, and energy equations for this quasi one-dimensional, steady,

isentropic flow can be expressed, respectively as

Continuity:

(1)

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Momentum:

(2)

Energy:

(3)

where subscripts 1 and 2 denote different locations along the nozzle. In addition, we have the perfect

gas equation of state,

(4)

As well as the relation for a calorically perfect gs,

(5)

equations (1) and (5) can be solved analytically for the flow through the nozzle.

Figure (c)

2.1 Assumed Model

Steady, quasi-one-dimensional. There are gradual variations in the geometry, so that the flow near the

nozzle walls is not strictly along the x-direction. However, the flow angularity is very small. The

variation in properties can be calculated assuming that the properties are constant in each cross-

section. The cross-section area, A, is a function of x alone. Thus, all properties are functions of x

alone.

A= A(x); u = u(x); T=T(x), p= p(x) etc.

Continuity:

, (6)

the mass flow rate, is constant.

Momentum: (no friction; differential form of the Euler equation)

(8)

Hence,

Using (8) in (10), (10)

. Isentropic process: . (11)

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Thus,

(12)

Also,

(13)

Case 1: M<1

dA, dp have the same sign. Thus, as A increases, p increases.

dA , du have opposite signs. Thus as A increases, u decreases.

Diverging duct in subsonic flow: pressure increases, speed decreases.

Converging duct in subsonic flow: pressure decreases, speed increases.

Case 2: M>1

dA, dp have opposite signs. Thus as A increases, p decreases.

dA, du have the same sign. Thus as A increases, u increases.

Diverging duct in supersonic flow: pressure decreases, speed increases.

Case 3: M = 1

dA/dx is 0. Thus we have either a maximum or minimum of area.

The maximum area case is not of much interest, since there is no way to reach Mach 1 at

this point, with flow from either direction.

So the case of interest is where the area becomes a minimum: a "throat".

From mass conservation, where the * denotes conditions at Mach 1

So,

(14)

(15)

(16)

(17)

(18)

(19)

(20)

Substitute into A/A*:

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(21)

Thus, for a given isentropic flow, i.e., a flow with mass flow rate, stagnation temperature and

stagnation pressure all fixed, there are two solutions for a given value of A/A*: One solution is

subsonic, the other is supersonic.

2.2 Mass Flow Rate Through a Nozzle

For given stagnation conditions are fixed.

(22)

For a given throat area, stagnation pressure and stagnation temperature, the maximum mass flow rate

is the value where the Mach number at the throat reaches 1.0. This is called the "choked mass flow

rate." To increase the mass flow rate, we have to increase the stagnation pressure, decrease the

stagnation temperature, or increase the throat area.

, or

(23)

(24)

For M=1, R=286.7 J/Kg K and γ = 1.4 for air,

(25)

Where is in , in Kelvin and in .

3. METHODOLOGY AND IMPLEMENTATION

3.1 Modeling the Nozzle

We have developed the model in such a way that it is

assumed to be in the mid-section of the nozzle. We used

GAMBIT modeling package for developing the wire frame

which resembles the cross-section of the rocket nozzle.

Figure.1 Nozzle Dimensions

3.2 Meshing in GAMBIT

In GAMBIT the model is meshed using Quad/Triangular elements and proper care is taken while

meshing the regions near the walls of the nozzle so as to get more refinement in that particular

regions. As any computational process requires a mesh to carry computation this step is a primary and

most important to start the problem. The mesh created in GAMBIT is as shown below figure.2

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We can see that the meshing near the

boundary of the nozzle is more refined

when compared to other regions of

mesh. After the completion of the

meshing we need to specify the

boundary types like the inlet, outlet,

walls, and continuum type.

Figure.2 Fine Mesh Near Walls

3.5 Boundary Conditions Used

1. Mass flow inlet

2. Outlet

3. Walls

Specification of the boundary zones has to be done in GAMBIT only, there is no possibility to specify

the boundary zones in FLUENT. So Proper care has to be taken while defining the boundary zones in

GAMBIT. With all the zones defined properly the mesh is exported to the solver. The solver used in

this problem is FLUENT. The exported mesh file is read in fluent for solving the problem.

3.3 Solving

Fluent analysis is carried out for nozzle at different Mach numbers and at different Nozzle pressure

ratios.

The steady axisymmetric implicit formulation with coupled solver with K-e turbulence model is

chosen. The mass flow of the burning solid propellant was modeled using a user-defined function

4. RESULTS

4.1 NOZZLE Analysis for Determination of Pressure,Velocity,Mass flow Rate and Forces

Fluent analysis is carried out at different Mach numbers and the values are tabulated at Different

Mach Numbers

Table 2

Mach No.

Maximum

Pressure

(atm)

Maximum

Velocity

(m/sec)

Maximum

Forces

(N)

Maximum

Flow rate

(kg/sec)

1 2.2426 1864.675 31748.542 -3.5058

1.2 2.24189 1866.671 32079.31 -3.5537

2.0 2.16971 1870.457 31807.389 -4.649

2.2 2.92254 1886.233 44094.323 -5.5084

3.0 9.6542 1883.28 33875.766 -4.488281

3.2 12.99903 1885.267 34029.145 -3.83

4.0 39.9912 1898.31 39506.234 -3.82

4.2 52.05184 1899.75 34653.321 -3.8249

5.0 85.0891 1891.65 44749.05 -4.32

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4.2 At Different Nozzle Pressure Ratios Table.3

NPR PRESSURE

(atm)

MAXIMUM

PRESSURE

(atm)

MAXIMUM

VELOCITY

(m/sec)

MAXIMUM

FORCE

(N)

MAXIMUM

FLOW RATE

(kg/sec)

1.2 0.313836 2.23954 1862.358 29407.048 -3.311035

1.29 0.3373 2.239229 1849.845 28884.084 -3.180176

1.34 0.3504 2.23906 1833.296 28588.05 -3.4653

1.44 0.37660 2.23874 1832.658 28006.213 -3.30957

1.72 0.4498 2.39809 1783.231 26356.765 -3.4146

1.97 0.51521 2.24297 1759.817 24974.712 -3.26953

2.09 0.5465 2.241855 1753.713 24565.12 -3.2363

2.26 0.59105 2.30605 1746.947 24236.067 -3.4619

4.3 The following figures illustrate Contours of Pressure, Velocity Magnitude, Temperature,

Turbulence, Mass Flow rates at Mach no:1

Figure.3 Contour of Static Pressure Figure.4 Mass Flow Rate on Mass Flow Inlet

Figure.5 Contours of Static Temperature Figure.6 Static Temperature Plot at Wall-1-A,B

Figure.7 Contour of Mach Number Figure.8 Wall Y-Star at Wall-1-A,B, Wall-3

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4.4 At Different NPR’s

Figure.9 Contour of Static Pressure at NPR 1.2 Figure.10 Contour of Static Pressure At NPR 1.79

Figure.11 Static Temperature Plot at NPR 1.2 Figure.12 Static Temperature Plot at NPR 1.97

Figure.13 Contour of Static Temperatue at NPR 1.2 Figure.14 Contour of Velocity Magnitude At NPR1.97

Figure.15 Wall Y-Star at NPR 1.97 Figure.16 Total Pressure at NPR 1.97

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5. CONCLUSIONS

Computer aided solutions are developed using Fluent Analysis. Solutions are evaluated at

different Mach numbers corresponding Mass flow rates, Maximum Velocity, Maximum

Pressure, Maximum force are determined. Mass Flow rates were found to decrease

monotonically with Mach numbers. Variation in static pressure increases with Mach number less

than 6 and was found to be identical from the inlet throat to the exit for Mach number values

higher than 6.With the corresponding increase of Mach number corresponding velocity also

increases. Maximum forces alternately increase and decrease with respect to Mach number.

Solutions are evaluated at different Nozzle pressure ratios(NPR) corresponding Mass flow rates,

Maximum Velocity, Maximum Pressure, Maximum force are determined. Mass Flow rates were

found to decrease monotonically with Nozzle pressure ratios(NPR).Variation in static pressure

increases with Nozzle pressure ratios(NPR).With the corresponding increase of Nozzle pressure

ratios(NPR) corresponding velocity decreases. Maximum forces and Maximum velocity

decrease with respect to Nozzle pressure ratios (NPR).

The mass flow of the burning solid propellant was modeled using a user-defined function. Both

turbulence and conjugate heat transfer were modeled in this problem. Finally it’s emphasized the

importance of checking other parameters (such as total mass balance) in addition to the residual

values.

REFERENCES

[1] Adamson, T.C., Jr., and Nicholls., J.A., “On the structure of jets from Highly underexpanded

Nozzles into Still Air,” Journal of the Aerospace Sciences, Vol.26, No.1, Jan 1959, pp. 16-24.

[2] Lewis, C. H., Jr., and Carlson, D. J., “Normal Shock Location in underexpanded Gas and Gas

Particle Jets,” AIAA Journal, Vol 2, No.4, April 1964, pp. 776-777.

[3] Romine, G. L., “Nozzle Flow separation,” AIAA Journal, Vol. 36, No.9, Sep. 1998. Pp 1618-

1625.

[4] Anderson Jr, J. D., “Computational Fluid Dynamics the basic with Applications,” McGraw-

Hill, revised edition 1995.

[5] Dutton, J.C., “Swirling Supersonic Nozzle Flow,” Journal of Propulsion and Power, vol.3, July

1987, pp. 342-349.

[6] Elements of Propulsion: Gas Turbines and Rockets ---- Jack D. Mattingly

[7] Introduction to CFD---- H K VERSTEEG &W MALALASEKERA

[8] Rocket and Spacecraft Propulsion Principles, Practice and New Developments (2nd ed.,

Springer, 2005) ---- Turner M.

[9] Rocket Propulsion Elements- 7th edition ---- GEORGE P. SUTTON; OSCAR BIBLARZ

[10] FLUID MECHANICS----A K MOHANTY