Independent Research Project

Independent Project on: 

CONVERGENT DIVERGENT NOZZLE

Aim: 2D simulation of flow inside the CD nozzle

Objectives:

• 2D simulation of CD nozzle flow
• Plot Pressure and Velocity (Mach No) graph and contours
• Showing the Sonic condition at neck region

Introduction:

Nozzles- 

Most modern passenger and military aircraft are powered by gas turbine engines, which are also called jet engines. There are several different types of gas turbine engines, but all turbine engines have some parts in common. All gas turbine engines have a nozzle to produce thrust, to conduct the exhaust gases back to the free stream, and to set the mass flow rate through the engine. The nozzle sits downstream of the power turbine.

A nozzle is a relatively simple device, just a specially shaped tube through which hot gases flow. However, the mathematics which describe the operation of the nozzle takes some careful thought. As shown above, nozzles come in a variety of shapes and sizes depending on the mission of the aircraft. Simple turbojets, and turboprops, often have a fixed geometry convergent nozzle as shown on the left of the figure. Turbofan engines often employ a co-annular nozzle as shown at the top left. The core flow exits the center nozzle while the fan flow exits the annular nozzle. Mixing of the two flows provides some thrust enhancement and these nozzles also tend to be quieter than convergent nozzles. Afterburning turbojets and turbofans require a variable geometry convergent-divergent - CD nozzle as shown on the left. In this nozzle, the flow first converges down to the minimum area or throat, then is expanded through the divergent section to the exit at the right. The flow is subsonic upstream of the throat, but supersonic downstream of the throat. The variable geometry causes these nozzles to be heavier than a fixed geometry nozzle, but variable geometry provides efficient engine operation over a wider airflow range than a simple fixed nozzle.

CD nozzle- 

Ramjets, scramjets, and rockets all use nozzles to accelerate hot exhaust to produce thrust as described by Newton's third law of motion. The amount of thrust produced by the engine depends on the mass flow rate through the engine, the exit velocity of the flow, and the pressure at the exit of the engine. The value of these three flow variables are all determined by the nozzle design.

A nozzle is a relatively simple device, just a specially shaped tube through which hot gases flow. Ramjets and rockets typically use a fixed convergent section followed by a fixed divergent section for the design of the nozzle. This nozzle configuration is called a convergent-divergent, or CD, nozzle. In a CD nozzle, the hot exhaust leaves the combustion chamber and converges down to the minimum area, or throat, of the nozzle. The throat size is chosen to choke the flow and set the mass flow rate through the system. The flow in the throat is sonic which means the Mach number = 1 in the throat. Downstream of the throat, the geometry diverges and the flow is isentropically expanded to a supersonic Mach number that depends on the area ratio of the exit to the throat. The expansion of a supersonic flow causes the static pressure and temperature to decrease from the throat to the exit, so the amount of the expansion also determines the exit pressure and temperature. The exit temperature determines the exit speed of sound, which determines the exit velocity. The exit velocity, pressure, and mass flow through the nozzle determines the amount of thrust produced by the nozzle.

The equations which explain and describe why a supersonic flow accelerates in the divergent section of the nozzle while a subsonic flow decelerates in a divergent duct. We begin with the conservation of mass equation:

mdot = r * V * A = constant

where mdot is the mass flow rate, r is the gas density, V is the gas velocity, and A is the cross-sectional flow area. If we differentiate this equation, we obtain:

V * A * dr + r * A * dV + r * V * dA = 0

divide by (r * V * A) to get:

dr / r + dV / V + dA / A = 0

Now we use the conservation of momentum equation:

r * V * dV = - dp

and an isentropic flow relation:

dp / p = gam * dr / r

where gam is the ratio of specific heats. This is Equation #10 on the page which contains the derivation of the isentropic flow relations We can use algebra on this equation to obtain:

dp = gam * p / r * dr

and use the equation of state

p / r = R * T

where R is the gas constant and T is temperature, to get:

dp = gam * R * T * dr

gam * R * T is the square of the speed of sound a:

dp = (a^2) * dr

Combining this equation for the change in pressure with the momentum equation we obtain:

r * V * dV = - (a^2) * dr

V / (a^2) * dV = - dr / r

- (M^2) * dV / V = dr / r

using the definition of the Mach number M = V / a. Now we substitute this value of (dr /r) into the mass flow equation to get:

- (M^2) * dV / V + dV / V + dA / A = 0

(1 - M^2) * dV / V = - dA / A

This equation tells us how the velocity V changes when the area A changes, and the results depend on the Mach number M of the flow. If the flow is subsonic then (M < 1) and the term multiplying the velocity change is positive (1 - M^2 > 0). An increase in the area (dA > 0 ) produces a negative increase (decrease) in the velocity (dV < 0). For our CD nozzle, if the flow in the throat is subsonic, the flow downstream of the throat will decelerate and stay subsonic. So if the converging section is too large and does not choke the flow in the throat, the exit velocity is very slow and doesn't produce much thrust. On the other hand, if the converging section is small enough so that the flow chokes in the throat, then a slight increase in area causes the flow to go supersonic. For a supersonic flow (M > 1) the term multiplying velocity change is negative (1 - M^2 < 0). Then an increase in the area (dA > 0) produces an increase in the velocity (dV > 0). This effect is exactly the opposite of what happens subsonically. Why the big difference? Because, to conserve mass in a supersonic (compressible) flow, both the density and the velocity are changing as we change the area. For subsonic (incompressible) flows, the density remains fairly constant, so the increase in area produces only a change in velocity. But in supersonic flows, there are two changes; the velocity and the density. The equation:

- (M^2) * dV / V = dr / r

tells us that for M > 1, the change in density is much greater than the change in velocity. To conserve both mass and momentum in a supersonic flow, the velocity increases and the density decreases as the area is increased.

Methodology:

Geometry:

Space claim-

Select xy plane and draw a rough skecth

Give constraints and dimension

Use pull tool and make a 2D geometry

Meshing:

In meshing first select edges and use sizing and divide through no of divisions

on generating mesh it is unstructured mesh but we make it structured mesh by using face meshing option

     

Details of mesh

Create named selection

Set up:

Pressure based solver -> Absolute velocity formulation -> Steady state solver ->planar 2D space

check Energy equation -> laminar viscous model

Create materials -> Density Ideal gas -> Viscosity sutherland

     

In ANSYS FLUENT, if you choose to define the density using the ideal gas law for an incompressible flow, the solver will compute the density as where, In this form, the density depends only on the operating pressure and not on the local relative pressure field.

According to Anderson (2006), “Sutherland’s law is accurate for air over a range of several thousand degrees and is certainly appropriate for hypersonic viscous-flow calculations”. In the crucial undergraduate textbook by Frank White, states that it is “adequate over a wide range of temperatures”

Apply Boundary condition

Inlet

Outlet

Wall

Initialize -> Standard initialization through inlet

Reports -> coefficient of lift and drag

XY plots -> Pressure and Mach no

    

    

Results:

Co efficient of lift

Co efficient of Drag

Pressure through the Nozzle

Mach number through the Nozzle

Pressure contour

Velocity streamlines

Conclusion:

• In Pressure graph we see that presure is high at inlet and it becomes low at outlet
• In Mach number graph we see Mach no < 1 at inlet region, Mach no = 1 at neck region and Mach no > 1 at outlet region i.e., Subsonic, Sonic and Supersonic regions respectively
• Velocity streamlines showsthe velocity flow through the Nozzle