Week 2 - Converging diverging nozzle

Aim

To study the variation of mach number for different altitude and also to study the variation of Mach number across the domain and at what distance from the inlet does the flow become supersonic and to compare with the analytical results.

Introduction

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 supersonic 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.

Working

Its operation relies on the different properties of gases flowing at subsonic, sonic, and supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is isentropic (gas entropy is nearly constant). In a subsonic flow sound will propagate through the gas. 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 backward through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0).

As the gas exits the throat the increase in area allows for it to undergo a Joule-Thompson expansion wherein the gas expands at supersonic speeds from high to low pressure pushing the velocity of the mass flow beyond sonic speed.

When comparing the general geometric shape of the nozzle between the rocket and the jet engine, it only looks different at first glance, when in fact is about the same essential facts are noticeable on the same geometric cross-sections - that the combustion chamber in the jet engine must have the same "throat" (narrowing) in the direction of the outlet of the gas jet, so that the turbine wheel of the first stage of the jet turbine is always positioned immediately behind that narrowing, while any on the further stages of the turbine are located at the larger outlet cross section of the nozzle, where the flow accelerates.

Procedure

Model 

All Dimensions are in meters

Mesh

Mesh size 4mm

Models used 

Solver - Density based (It is used because the flow is compressible)

Velocity formulation -  Absolute

Time - Steady

2D space - Axisymmetric

Energy - On (Because we need to calculate temperature across the domain)

Viscous - k-epsilon - Standard - Standard wall functions 

Boundary conditions

Inlet - Pressure Inlet (Press - 101325 pa & Temp - 300K)

Outlet - Pressure Outlet (Press - 2735.775 pa)

Results

Mach number contour

Pressure contour

Temperature contour

Velocity contour

Plots

Velocity

Temperature

Pressure

Mach number = 1 at 0.53m (total lenght of the nozzle is 1 meter ), which is close to the throat section.

The nozzle can we divide into two section the first section is the converging section in that as the area decreases the mach number increases as the flow is subsonic and the second section is the diverging section. In the diverging section as the area increases the mach number increases as the flow starts to change fom sonic to supersonic.