DESIGN AND CFD ANALYSIS OF CONVERGENT AND DIVERGENT NOZZLE

Transcription

1 DESIGN AND CFD ANALYSIS OF CONVERGENT AND DIVERGENT NOZZLE 1 P.VINOD KUMAR, 2 B.KISHORE KUMAR 1 PG Scholar, Department ofmech,nalanda INSTITUTION OF ENGINEERING AND TECHNOLOGY KantepudiSattenapalli, GUNTUR,A.P, India, Pin: Assistant Professor, Department of MECH,NALANDA INSTITUTION OF ENGINEERING AND TECHNOLOGY,Kantepudi,Sattenapalli, GUNTUR,A.P, India, Pin: Abstract Nozzle is a device designed to control the rate of flow, speed, direction, mass, shape, and/or the pressure of the Fluid that exhaust from them. Convergent-divergent nozzle is the most commonly used nozzle since in using it the propellant can be heated incombustion chamber. In this project we designed a new Tri-nozzle to increase the velocity of fluids flowing through it. It is designed based on narrowed, increasing the speed of the jet to the speed of sound, and then expanded again. Above the speed of sound (but not below it) this expansion caused a further increase in the speed of the jet and led to a very efficient conversion of heat energy to motion. The theory of air resistance was first proposed by Sir Isaac Newton in According to him, an aerodynamic force depends on the density and velocity of the fluid, and the shape and the size of the basic convergent-divergent nozzle to have same displacing object. Newton s theory was soon throat area, length, convergent angle and divergent angle as single nozzle. But the design of Tri-nozzle is optimized to have high expansion co-efficient than single nozzle without altering the divergent angle. In the present paper, flow through thetri-nozzle and convergent divergent nozzle study is carried out by using SOLID WORKS PREMIUM 2014.The nozzle geometry modeling and mesh generation has been followed by other theoretical solution of fluid motion problems. All these were restricted to flow under idealized conditions, i.e. air was assumed to posses constant density and to move in response to pressure and inertia. Nowadays steam turbines are the preferredpower source of electric power stations and large ships, although they usually have a different design-to make best use of the fast steam jet, de done using SOLID WORKS CFD Software. Laval s turbine had to run at an impractically high Computational results are in goodacceptance with the experimental results taken from the literature. speed. But for rockets the de Laval nozzle was just what was needed. 1. Introduction to nozzle A nozzle is a device designed to control the Swedish engineer of French descent who, in trying to develop a more efficient steam engine, designed a turbine that was turned by jets of steam. The critical component the one in which heat energy of the hot high-pressure steam from the boiler was converted into kinetic energy was the nozzle from which the jet blew onto the wheel. De Laval found that the most efficient conversion occurred when the nozzle first direction or characteristics of a fluid flow (especially to increase velocity) as it exits (or enters) an enclosed chamber. A nozzle is often a pipe or tube of varying cross sectional area and it can be used to direct or modify the flow of a fluid (liquid or gas). Nozzles are frequently used to control the rate of flow, speed, direction, mass, shape, and/or the pressure of the stream that emerges from them. A jet exhaust produces a net thrust from the energy obtained from

2 combusting fuel which is added to the inducted air. This hot air is passed through a high speed nozzle, a propelling nozzle which enormously increases its kinetic energy. The goal of nozzle is to increase the kinetic energy of the flowing medium at the expense of its pressure and internal energy. Nozzles can be described as convergent (narrowing down from a wide diameter to a smaller diameter in the direction of the flow) or divergent (expanding from a smaller diameter to a larger one). A de Laval nozzle has a convergent section followed by a divergent section and is often called a convergent-divergent nozzle ("con-di nozzle"). Convergent nozzles accelerate subsonic fluids. If the nozzle pressure ratio is high enough the flow will reach sonic velocity at the narrowest point (i.e. the nozzle throat). In this situation, the nozzle is said to be choked. Increasing the nozzle pressure ratio further will not increase the throat Mach number beyond unity. Downstream (i.e. external to the nozzle) the flow is free to expand to supersonic velocities. Note that the Mach 1 can be a very high speed for a hot gas; since the speed of sound varies as the square root of absolute temperature. Thus the speed reached at a nozzle throat can be far higher than the speed of sound at sea level. This fact is used extensively in rocketry where hypersonic flows are required, and where propellant mixtures are deliberately chosen to further increase the sonic speed. Divergent nozzles slow fluids, if the flow is subsonic, but accelerate sonic or supersonic fluids. Convergent-divergent nozzles can therefore accelerate fluids that have choked in the convergent section to supersonic speeds. This CD process is more efficient than allowing a convergent nozzle to expand supersonically externally. The shape of the divergent section also ensures that the direction of the escaping gases is directly backwards, as any sideways component would not contribute to thrust. 2. Literature review 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 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 increases 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 attain just sonic speeds at the throat, pressure ratio to e maintained is (Pthroat / P inlet)=0.528.

3 Table1: speeds vs mach number 3.1 Conical Nozzles Reg Subs Trans So Super Hyper High- ime onic onic nic sonic sonic hyper Ma < sonic ch >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 The governing continuity, momentum, and energy equations for this quasi one-dimensional, steady, isentropic flow can be expressed, respectively 3. Types of nozzles Types of nozzles are several types. They could be based on either speed or shape. a. Based on speed The basic types of nozzles can be differentiated as Spray nozzles Ramjet nozzles b. Based on shape The basic types of nozzles can be differentiated as Conical Bell Annular Fig3.1 conical nozzles 1. Used in early rocket applications because of simplicity and ease of construction. 2. Cone gets its name from the fact that the walls diverge at a constant angle 3. A small angle produces greater thrust, because it maximizes the axial component of exit velocity and produces a high specific impulse 4. Penalty is longer and heavier nozzle that is more complex to build 5. At the other extreme, size and weight are minimized by a large nozzle wall angle Large angles reduce performance at low altitude because high ambient pressure causes overexpansion and flow separation 6. Primary Metric of Characterization: Divergence Loss 3.2 BELL and Dual Bell Fig3.2 (1): BELL and Dual Bell This nozzle concept was studied at the Jet Propulsion Laboratory in In the late 1960s, Rocket dyne

4 patented this nozzle concept, which has received attention in recent years in the U.S. and Europe. The design of this nozzle concept with its typical inner base nozzle, the wall in section, and the outer nozzle extension can be seen. This nozzle concept offers an altitude adaptation achieved only by nozzle wall in section. In flow altitudes, controlled and symmetrical flow separation occurs at this wall in section, which results in a lower effective area ratio. For higher altitudes, the nozzle flow is attached to the wall until the exit plane, and the full geometrical area ratio is used. Because of the higher area ratio, an improved vacuum performance is achieved. However, additional performance losses are induced in dualbell nozzles. Fig 3.2(2) performance losses are induced in dualbell nozzles. 3.3 Functions of Nozzle The purpose of the exhaust nozzle is to increase the velocity of the exhaust gas before discharge from the nozzle and to collect and straighten the gas flow. For large values of thrust, the kinetic energy of the exhaust gas must be high, which implies a high exhaust velocity. The pressure ratio across the nozzle controls the expansion process and the maximum uninstalled thrust for a given engine is obtained when the exit pressure (Pe) equals the ambient pressure (P0).The functions of the nozzle may be summarized by the following list: 1. Accelerate the flow to a high velocity with minimum total pressure loss. 2. Match exit and atmospheric pressure as closely as desired. 3. Permit afterburner operation without affecting main engine operation requires variable area nozzle. 4. Allow for cooling of walls if necessary. throat 5. Mix core and bypass streams of turbofan if necessary. 6. Allow for thrust reversing if desired. 7. Suppress jet noise, radar reflection, and infrared radiation (IR) if desired. 8. Two-dimensional and axisymmetric nozzles, thrust vector control if desired. 9. Do all of the above with minimal cost, weight, and boat tail drag while meeting life and reliability goals Introduction to convergent and divergent nozzle A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a hot, pressurized gas passing through it to a higher speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines and rocket engine nozzles. It also sees use in supersonic jet engines. Similar flow properties have been applied to jet streams within astrophysics. Fig4: convergent and divergent nozzle 5. Conditions for operation

5 A de Laval nozzle will only choke at the throat if the pressure and mass flow through the nozzle is sufficient to reach sonic speeds, otherwise no supersonic flow is achieved, and it will act as a Venturi tube; this requires the entry pressure to the nozzle to be significantly above ambient at all times (equivalently, the stagnation pressure of the jet must be above ambient). In addition, the pressure of the gas at the exit of the expansion portion of the exhaust of a nozzle must not be too low. Because pressure cannot travel upstream through the supersonic flow, the exit pressure can be significantly below the ambient pressure into which it exhausts, but if it is too far below ambient, then the flow will cease to be supersonic, or the flow will separate within the expansion portion of the nozzle, forming an unstable jet that may "flop" around within the nozzle, producing a lateral thrust and possibly damaging it. In practice, ambient pressure must be no higher than roughly 2 3 times the pressure in the supersonic gas at the exit for supersonic flow to leave the nozzle. Fig5: mach number condition Its operation relies on the different properties of gases flowing at subsonic and supersonic speeds. Thespeed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flowthrough a de Laval nozzle is isentropic (gas entropy is nearly constant). In a subsonic flow the gas is compressible, and sound will propagate through it. At the "throat", where the cross-sectional area is at its minimum, the gas velocity locally becomes sonic (Mach number = 1.0), a condition called choked flow. As the nozzle cross-sectional area increases, the gas begins to expand, and the gas flow increases to supersonic velocities, where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0). 5.1 Fluid flow inside convergent and divergent nozzle A converging-diverging nozzle ('condi' nozzle, or CD-nozzle) must have a smooth area law, with a smooth throat, da/dx=0, for the flow to remain attached to the walls. The flow starts from rest and accelerates subsonically to a maximum speed at the throat, where it may arrive at M<1 or at M=1, as for converging nozzles. Again, for the entry conditions we use 'c' (for chamber) or 't' (for total), we use 'e' for the exit conditions, and '*' for the throat conditions when it is choked (M * =1). If the flow is subsonic at the throat, it is subsonic all along the nozzle, and exit pressure pe naturally adapts to environmental pressure p0 because pressure-waves travel upstream faster (at the speed of sound) than the flow (subsonic), so that pe/p0=1. But now the minimum exit pressure for subsonic flow is no longer pe=pt(2/(γ+1)) γ/(γ 1) (pe/p0=0.53 for γ=1.4), since the choking does not take place at the exit but at the throat, i.e. it is the throat condition that remains valid, p * =pt(2/(γ+1)) γ/(γ 1), e.g. p * /p0=0.53 for γ=1.4; now the limit for subsonic flow is pe,min,sub>p * because of the pressure recovery in the diverging part.

6 However, if the flow is isentropic all along the nozzle, be it fully subsonic or supersonic from the throat, the isentropic equations apply But if the flow gets sonic at the throat, several downstream conditions may appear. The control parameter is discharge pressure, p0. Let consider a fix-geometry CD-nozzle, discharging a given gas from a reservoir with constant conditions (pt,tt). When lowering the environmental pressure, p0, from the no flow conditions, p0=pt, we may have the following flow regimes (a plot of pressurevariation along the nozzle is sketched in Fig. 2): Subsonic throat, implying subsonic flow all along to the exit (evolution a in Fig 2). Sonic throat (no further increase in mass-flow-rate whatever low the discharge pressure let be). Flow becomes supersonic after the throat, but, before exit, a normal shockwave causes a sudden transition to subsonic flow (evolution c). It may happen that the flow detaches from the wall (see the corresponding sketch). o o Flow becomes supersonic after the throat, with the normal shockwave just at the exit section (evolution d). Flow becomes supersonic after the throat, and remains supersonic until de exit, but there, three cases may be distinguished: Oblique shock-waves appear at the exit, to compress the exhaust to the higher back pressure (evolution e). The types of flow with shock-waves (c, d and e in Fig. 2) are named 'over-expanded' because the supersonic flow in the diverging part of the nozzle has lowered pressure so much that a recompression is required to match the discharge pressure. That is the normal situation for a nozzle working at low altitudes (assuming it is adapted at higher altitudes); it also occurs at short times after ignition, when chamber pressure is not high enough. Adapted nozzle, where exit pressure equals discharge pressure (evolution f). Notice that, as exit pressure pe only depends on chamber conditions for a choked nozzle, a fixgeometry nozzle can only work adapted at a certain altitude (such that p0(z)=pe). Expansion waves appear at the exit, to expand the exhaust to the lower back pressure (evolution e); this is the normal situation for nozzles working under vacuum. This type of flow is named 'under-expanded' because exit pressure is not low-enough, and additional expansion takes place after exhaust. 5.2 Choked flow Chokingisa compressible flow effect that obstructs the flow, setting a limit to fluid velocity because theflow becomes supersonic and perturbations cannot move upstream; in gas flow, choking takes place when a subsonic flow reaches M=1, whereas in liquid flow,chokingtakes place when an almost incompressible flow reaches the vapour pressure (of the main liquid or of a solute), and bubbles appear, with the flow suddenly jumping to M>1 Going on with gas flow and leaving liquid flow aside, we may notice that M=1 can only occur in a nozzle neck, either in a smooth throat where da=0, or in a singular throat with discontinuous area slope (a kink in nozzle profile, or the end of a nozzle). Naming with a '*' variables the stage where M=1 (i.e. the sonic section, which may be a real throat within

7 the nozzle or at some extrapolated imaginary throat downstream of a subsonic nozzle. 5.3 Area ratio Nozzle area ratio ε (or nozzle expansion ratio) is defined as nozzle exit area divided by throat area, ε Ae/A *, in converging-diverging nozzles, or divided by entry area in converging nozzles. Notice that ε sodefined is ε>1, but sometimes the inverse is also named 'area ratio' (this contraction area ratio is bounded between 0 and 1); however, although no confusion is possible when quoting a value (if it is >1 refers to Ae/A *, and if it is <1 refers to A * /Ae), one must be explicit when saying 'increasing area ratio' (we keep toε Ae/A * >1). To see the effect of area ratio on Mach number, (14) is plotted in Fig. 1 for ideal monoatomic (γ=5/3), diatomic (γ=7/5=1.40), and low-gamma gases as those of hot rocket exhaust (γ=1.20); gases like CO2 and H2O have intermediate values (γ=1.3). Notice that, to get the same high Mach number, e.g. M=3, the area ratio needed is A * /A=0.33 for γ=1.67 and A * /A=0.15 for γ=1.20, i.e. more than double exit area for the same throat area (that is why supersonic wind tunnels often use a monoatomic working gas. expansion of steam in nozzle neither heat is supplied or rejected work. As steam passes through the nozzle it loses its pressure as well as heat. The work done is equal to the adiabatic heat drop which in turn is equal to Rankine area. 6.2 Velocity of Steam: Steam Enters nozzle with high pressure and very low velocity (velocity is generally neglected). Leaves nozzle with high velocity & low pressure All this is due to the reason that heat energy at steam is converted into K.E as it passes through nozzle. The final or outlet velocity at steam can be found as follows, Let C-Velocity of steam at section considered (m/sec) h - enthalpy at steam at inlet h - enthalpy at steam at outlet h - heat drop during expansion at steam (h h ) (for 1 kg of steam) Gain in K.E. = adiabatic heat drop = h C= (2 * 1000*h ) = h In practice there is loss due to friction in the nozzle and its value from 10-15% at total heat drop. Due to this the total heat drop is minimized. Let heat drop after reducing friction loss be kh Ratio A * /A (i.e. throat area divided by local area) vs. Mach number M, for γ=1.20 (beige), γ=1.40 (green), and γ=1.67] (red). 6. THEORETICAL BACK GROUND 6.1 Flows through Nozzles: The steam flow through the nozzles may be assumed as adiabatic flow. Since during the Velocity (C) = kh 6.3 Discharge Through The Nozzle And Condition For Its Maximum Value: p - initial pressure at steam v - initial volume at 1 kg of steam at P (m ) p -steam pressure at throat v - volume at 1kg steam at P (m ) A-area at cross section at nozzle at throat (m ) C- velocity at steam (m/s)

8 Steam passing through nozzle follows adiabatic process in which pv = constant n= for saturated steam n= 1.3 for super saturated steam for wet steam the value at n can be calculated by Dr.Zenner s equation n= x x- dryness fraction at steam (initial) workdone per 1 kg at steam during the cycle (Rankine cycle) W = (p v - p v ) Already we know Gain in K.E = adiabatic heat drop Per 1 kg = (p v - p v ) = workdone during Ranlinecycl = p v (1- ).(1) Also p v = p v = ( v v ) => = ( p p ) (2) =>v = v ( p p )..(3) equation 2 in 1 = p v (1- ) = p v [1- ( p p ) ] = p v [1- = p v [1- ( p p ( p p ) ] ) ] c = 2 { p v [1- ( p p ) ]} C = (2{ n p n1 1v 1 [1- ( p 2 p1 ) n1 n ]} If m is the mass of steam discharged in kg/sec m = by substituting value of c &v we get m = ( ) ( 2{ p v [1- ( p p ) ]}) m = (2{ p v [( p p ) - ( p p ) ]}) it is obvious form above equation that there is only one value at the ratio (critical pressure ratio) p p which will produce max discharge and this can be obtained by differenciating m with respect to p p and equating to zero and other quantities except p p remains here constant => [(p p ) - ( p p ) ] = 0 => [ (p p ) - ( )(p p ) ] =>( p p ) = ( ) ( p p ) =>( p p ) = ( 2 ) p p = ( 2 ) p p = ( 2 ) Hence discharge through the nozzle will be maximum when critical pressure ratio is = p p = ( 2 ) By substituting p p value in mass equation we get the maximum discharge m = (2{ p v [( p p ) - ( p p ) ]})

9 = (2{ p v [[ ( 2 [( 2 ) ] ]}) = [( 2 ) ]}) = [( 2 ) 1]}) = [( 2 ) 1]]}) = [( 1)] }) ( )]}) m = = (2{ p v [[ ( 2 (2{ p v [[ ( 2 (2{ p v [[ ( 2 (2{ p v [ ( 2 (2{ p v [ ( 2 A n ( ) [( 2 ) ) ) ] ) ) ) ) ) By substituting p p = ( 2 in equation c we get C C = 2( ) p v [1- (( 2 = 2( ) p v [1- ] = 2( ) p v ( ) C = 2( ) p v ) ) From maximum equation, it is evident that the maximum mass flow depends only on the inlet conditions ( p v ) and the throat area.and it is independent at final pressure at steam i.e., exit at the nozzle Note: pv = constant = constant = constant p p = ( ) = ( ) Specific volume v = v ( p p ) Apparent temperature T = T ( p p ) Nozzle efficiency: When the steam flows through a nozzle the final velocity of steam for a given pressure drop is reduced due to following reasons I. The friction between the nozzle surface and II. III. steam Internal friction of steam itself and The shock losses Most of these frictional losses occur between the throat and exit in convergent divergent nozzle. These frictional losses entail the following effects. 1. The expansion is no more isentropic and enthalpy drop is reduced. 2. The final dryness fraction at steam is increased as the kinetic energy gets converted into heat due to friction and is observed by steam. 3. The specific volume of steam is increased as the steam becomes more dry due to this frictional reheating 1 K = = actual = isentropic

10 Nozzle efficiency is the ratio of actual enthalpy drp to the isentropyenthalpy drop between the same pressure. Nozzle efficiency = If actual velocity at exit from the nozzle is c and the velocity at exit when the flow is isentropic is c then using steady flow energy equation. In each case we have h + h + = h + =>h - h = = h + =>h - h = Nozzle efficiency = Inlet velocity c is negligibly small Nozzle efficiency = Sometimes velocity coefficient is defined as the ratio of actual exit velocity to the exit velocity when the flow is isentropic between the same pressures. i.e., velocity coefficient = Velocity coefficient is the square root of the nozzle efficiency when the inlet velocity is assumed to be negligible. Enthalpy drop = (( ) p v [1- ( p p ) ]) = ( p p ) c = {2( ) p v [1- ( p p ) ]} It is an easy-to-learn tool which makes it possible for mechanical designers to quickly sketch ideas, experiment with features and dimensions, and produce models and detailed drawings. A Solid Works model consists of parts, assemblies, and drawings. Typically, we begin with a sketch, create a base feature, and then add more features to the model. (One can also begin with an imported surface or solid geometry). We are free to refine our design by adding, changing, or reordering features. Associatively between parts, assemblies, and drawings assures that changes made to one view are automatically made to all other views. We can generate drawings or assemblies at any time in the design process. The Solid works software lets us customize functionality to suit our needs. 8. Modeling of convergent divergent nozzle First select a new file and front plane Draw sketch as follows v = v ( p p ) T = T ( p p ) A = m = 7. SOLID WORKS Solid Works is mechanical design automation software that takes advantage of the familiar Microsoft Windows graphical user interface. Then go to features and make revolve

11 Table2: List of different general settings in SolidWorks Flow Simulation Now flow simulation Wizard 3d model of c & d nozzle 9. FLOW SIMULATION SolidWorks Flow Simulation 2010 is a fluid flow analysis add-in package that is available forsolidworks in order to obtain solutions to the full Navier-Stokes equations that govem the motion of fluids. Other packages that can be added to SolidWorks include SolidWorks Motion and SolidWorks Simulation. A fluid flow analysis using Flow Simulation involves a number of basic steps that are shown in the following flowchart in figure. Set units Flow type- internal in x-axis direction Figure: Flowchart for fluid flow analysis using Solidworks Flow Simulation General setting Next gases add air as fluid

12 Computational domain Mach number 9.1 For 5 bar inlet pressure Boundary conditions Inlet mass flow rate 50 kg/sec, pressure 5 bar (select inside faces) Goals result table Result Pressure 9.2 For pressure 10bars Boundary conditions Give mass flow rate 50kg/s and pressure 10bars Pressure Velocity Velocity

13 Mach number Mach number Goals tables bar Pressure Velocity 10. Graphs bar Pressure Mach number Velocity Table: Results Given Pressure Pressure Velocity Mach number 5bars bars

14 11. Conclusion: Modeling and analysis of Convergent and divergent nozzle is done in Solidworks 2016 Modeling of single nozzle is done by using various commands in solid works and analyzed at various pressures i.e., at 5bars and 10bars respectively. Analysis is done on single nozzle at 5 bars and 10 bars and values are noted. Velocity of nozzle at 5 bar and 10 bar pressure are tabulated in results table. Thus variations in velocities at certain given pressures of convergent and divergent nozzle are analyzed in this project. 12. References: A.A.Khan and T.R.Shembharkar, Viscous flow analysis in a Convergent-Divergent nozzle. Proceedings of the international conferece on Aero Space Science and Technology, Bangalore, India, June 26-28, H.K.Versteeg and W.MalalaSekhara, An introduction to Computational fluid Dynamics, British Library cataloguing pub, 4th edition, David C.Wil Cox, Turbulence modeling for CFD Second Edition S.Majumdar and B.N.Rajani, Grid generation for Arbitrary 3-D configuration using a Differential Algebraic Hybrid Method, CTFD Division, NAL, Bangalore, April Layton, W.Sahin and Volker.J, A problem solving approach using Les for a backward facing-step M.M.Atha vale and H.Q. Yang, Coupled field thermal structural simulations in Micro Valves and Micro channels CFD Research Corporation. Lars Davidson, An introduction to turbulence Models, Department of thermo and fluid dynamics, Chalmers university of technology, Goteborg, Sweden, November, Kazuhiro Nakahashi, Navier-Stokes Computations of two and three dimensional cascade flow fields, Vol.5, No.3, May-June 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 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 Romine, G. L., Nozzle Flow separation, AIAA Journal, Vol. 36, No.9, Sep Pp Anderson Jr, J. D., Computational Fluid Dynamics the basic with Applications, McGrawHill, revised edition Dutton, J.C., Swirling Supersonic Nozzle Flow, Journal of Propulsion and Power, vol.3, July 1987, pp Elements of Propulsion: Gas Turbines and Rockets ---- Jack D. Mattingly Introduction to CFD---- H K VERSTEEG &W MALALASEKERA