COMPUTATIONAL ANALYSIS OF DE LAVAL NOZZLE PDF

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COMPUTATIONAL ANALYSIS OF DE LAVAL NOZZLE ANANDRAJA PERUMAL Beihang University of Aeronautics & Astronautics Beijing, China * Corresponding Author: [email protected] ABSTRACT This CFD analysis is to demonstrate the computational analysis of physical flow phenomena involved in a nozzle of roc...

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COMPUTATIONAL ANALYSIS OF DE LAVAL NOZZLE ANANDRAJA PERUMAL Beihang University of Aeronautics & Astronautics Beijing, China *

Corresponding Author: [email protected]

ABSTRACT This CFD analysis is to demonstrate the computational analysis of physical flow phenomena involved in a nozzle of rocket engine. The nozzle is assumed to be a De Laval nozzle design. The grid has been generated in Gambit and Euler or N-S equations based computation has been done in the Fluent. For post processing Tec plot has been used. Both inviscid and viscous cases have been simulated for a comparative study. INTRODUCTION The nozzle of a rocket engine is a carefully shaped aft portion of the thrust chamber that controls the expansion of the exhaust gas so that the thermal energy of combustion is effectively converted into kinetic energy in order to propel the rocket. So, it changes the low velocity, high pressure and temperature gas flow into a high velocity, low pressure and temperature flow. Typical nozzle geometry includes a converging section, throat and a diverging section as shown in the Fig. 1.

Convergent region serves as a transition between the chamber case and the throat region. Throat region is the smallest cross-sectional area than the rest of the engine. Exhaust gases from combustion are pushed into throat region of nozzle and are compressed to high pressure. In the divergent section, there is a gradual increase in the cross-sectional area allowing gases to expand and push against walls creating thrust. Mathematically, ultimate purpose of nozzle is to expand gases as efficiently as possible so as to maximize exit velocity. The nozzle exit velocity that can be achieved is governed by the nozzle expansion ratio ε, defined as the ratio between the nozzle exit area and throat area



Aexit A  e* Athroat A

In addition to the momentum thrust, there is an additional pressure thrust because of the pressure forces difference at the nozzle exit. Combining the momentum and pressure thrust, the total thrust (F) produced by the rocket nozzle can be expressed as

 ve   pe  pa Ae F m

Where pe and Ae are the pressure and cross section area at the nozzle exit, and pa is the ambient pressure.

LAVAL NOZZLE 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 supersonic speed, and upon expansion, to shape the exhaust flow so that the heat energy propelling the flow is maximally converted into directed kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines, and is used as a rocket engine nozzle. It also sees use in supersonic jet engines. The Laval nozzle was developed by Swedish inventor Gustaf de Laval in 1888 for use on a steam turbine. This principle was first used in a rocket engine by Robert Goddard. Very nearly all modern rocket engines that employ hot gas combustion use de Laval nozzles. COMPUTATIONAL FLUID DYNAMICS CFD is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyse problems that involve fluid flows. The fundamental basis of any CFD problem is the Navier-Stokes equations, which define any fluid flow. These equations can be simplified by removing terms describing viscosity to yield the Euler equations. CFD gives you the power to simulate flows, heat and mass transfer, moving bodies, etc., through computer modelling. The CFD began in the 60’s in the aerospace industry, and nowadays has become into a vital tool for many industries for the prediction of fluid flow. It has been expanded significantly to different industrial applications and industrial processes involving heat transfer, chemical reactions, two-phase flow, phase changes and mass transfer, among others. In order to obtain an approximate solution numerically, a discretization method have to be used to approximate the differential equations by a system of algebraic equations, which can later be solved with the help of a computer. The approximations are applied to small domains in time and/or space. The accuracy of numerical solutions depends on the quality of the discretization used. It is important to bear in mind that numerical results are always approximate. Discretization errors can be reduced by using more accurate interpolation or approximations or by applying the approximations to smaller regions, but this usually increases the time and cost of obtaining the solution. Compromise is needed in solving the discretized equations. Direct solvers, which obtain accurate solutions, are not very used because they are too expensive. Otherwise iterative methods are more. PROBLEM DESCRIPTION A computational investigation of a Laval nozzle has been made in the present study utilizing commercially available computational fluid dynamics code FLUENT-13®. The meshing has been done in GAMBIT®. For efficient computation of results the nozzle is modelled as axis-symmetric. The applied boundary conditions are listed in Table. Total temperature in combustion chamber (Tc) K

Total pressure in combustion chamber (Pc) MPa

Operating ambient pressure (Pa) MPa

Molecular weight (M) Kg/KgMol

Specific heat at constant pressure (Cp) J/KgK

3720

9.5

0.1

28.531

3522.3

GRID GENERATION IN GAMBIT Gambit is software used for geometry and mesh generation. The Laval nozzle geometry has been created in the Gambit software. A vertex is the lowest order entity in the gambit. These are joined by edges which is the next higher geometric entity of Gambit, in order to give a closed profile. Then these edges are combined to form a face. Face is the highest order geometric entity in two dimensional modelling. Below Figure shows the stepwise geometry modelling procedure.

In order to better capture the boundary layer a boundary layer mesh is created along the wall of the nozzle and axis of the nozzle, a growth factor of 1.05 and 1.03 and with 120 and 40 no. of rows. Edges are then discretized in order to generate a domain mesh. Quad elements are used to generate a mapped domain mesh in below figure. In Gambit the edges are also assigned with boundary types. The boundary types which are assigned to different edges are depicted in Table. After assigning the boundary conditions the 2-D mesh file is exported for Fluent with .msh extension. TABLE: MESHING DETAILS Meshing

Edge # 1 2 3 4 5 6 7 8

AREA Combustion chamber Combustion chamber Throat area Throat area Exit area Exit area Inlet area Outlet area

Ratio 1.3 1.3 1.3 1.3 1.3 1.3 1.5 1.5

Interval Count 40 40 40 40 40 40 120 120

Table: boundary conditions BOUNDARY NAME

BOUNDARY TYPE

EDGES

Pressure inlet

PRESSURE INLET

EDGE 9

Pressure-outlet

PRESSURE OUTLET

EDGE 6

AXIS

EDGE1,EDGE 2, EDGE 3, EDGE4, EDGE 5

WALL

EDGE7, EDGE8, EDGE10,EDGE15,EDGE16

axis

wall

MESH

FLUENT Fluent is a package of simulation computational fluid dynamics (CFD) and the most used in the world. The structure of Fluent allows you to incorporate a lot of models for different physical and chemical processes. Not only can you perform simulations of laminar or turbulent flow, Newtonian or non-Newtonian, compressible or incompressible, single phase or multiphase, but also processes of heat transfer by radiation, conduction and convection of course, as well as melting processes and chemical reactions such as burning of gases, liquids and solid fuels.

The step wise procedure to be followed in Fluent is shown in the Fig.5. STEP 1

•GRID CHECK - Performed in order to ensure absence of negative volumes

STEP 2

•SCALE - The unit "mm" is selected in mesh was created in sub menu in the scale option

STEP 3

•SOLVER - Density based, steady,implict, axisymmetric

STEP 4

•Energy - select the energy equation

STEP 5

•Model - viscous- select the K-epsilon 2 equation and spalart - allmaras 1 equation

STEP 5

•MATERIALS - Ideal gas in density & Sutherland law is selected in viscossity

STEP 6

•OPERATING CONDITIONS - Set zero

STEP 7

•BOUNDARY CONDITIONS - Entered from the data in Table 1

STEP 8

•SOLUTION - Enter Courant number equals to 0.1 and first order upwind

STEP 9

•INITIALIZATION - Initialize from inlet

STEP 10

•RUN CALCULATION - Give 100000 iteration

Another analysis would be run without viscous effects. This is achieved by selecting inviscid model instead of Spalart-Allmaras turbulence model or K-epsilon 2equation or turbulent model. After the solution is converged for both the cases, the post processing would be done in Tecplot-2010®. RESULTS Different flow parameters distribution along the nozzle axis is plotted in Figs.6-11. The upper half nozzle contours are for the case with K-epsilon 2 equation turbulence model. While the lower half nozzle contours are for the inviscid case.

FIGURE: STATIC PRESSURE

FIGURE: DYNAMIC PRESSURE

FIGURE: DENSITY

FIGURE: AXIAL VELOCITY

FIGURE: RADIAL VELOCITY

FIGURE: MACH NUMBER

FIGURE: STATIC TEMPERATURE

FIGURE: TOTAL ENERGY

FIGURE: ENTHALPHY

CONCLUSION A successful CFD investigation of De Laval rocket nozzle has been done. For a comparative study both turbulent and inviscid flows have been analysed. It is evident from the result plots that for the same grid the inviscid case is not capturing any boundary layer while in the viscous case a clear layer has been generated.

References [1]

John D. Anderson, Jr. Computational Fluid Dynamics the Basics with Applications McGraw-Hill Company (Original version), 2002.

[2]

Bengt Andersson, Ronnie Andersson. Computational Fluid Dynamics for Engineers Cambridge University press, (2005).

[3]

Fluent 12.1 Help and GAMBIT 6.0 Help and online research papers....