Simulation on Compressible Flow – Transonic Flow in A Converging/Diverging Nozzle

 

Figure above shows the velocity magnitude of a transonic flow in a converging/diverging nozzle. The inlet velocity is the bright blue color. From the color, we can see that the velocity from the (left to the right) along the nozzle increases although the middle of the nozzle narrows.

From the Ref (1), it’s has been derivate an equation for compressible flow in diffuser, nozzle and wind tunel, i.e.

dA/A = (M²-1) du/u

Where:

 A = cross sectional area

M = Mach number

u = velocity in x direction

This equation tells us  four information:

1)      For Mach number between 0 and 1 (also known as subsonic flow), the quantity in parentheses is negative. Hence, an increase in velocity (positive du) is associated with a decrease in area (negative dA). Likewise, a decrease in velocity (negative du) is associated with an increase in area (positive dA). These results are similar to the familiar trends for incompressible flow. We see that subsonic compressible flow is qualitatively (but not quantitatively) similar to incompressible flow.

2)      For M larger than 1 (also known as supersonic flow), the quantity in parenthesis is positive. Hence, the increase in velocity (positive du) is associated with an increase in area (positive dA). Likewise, a decrease in velocity (negative du) is associated with a decrease in area (negative dA).

3)      For M equal to 1 (also known as sonic flow). This equation shows that dA = 0 even though a finite du exists.

4)      For M equal to 0, then we have dA/A = -du/u. This is the familiar continuity equation for incompressible flow.

The above picture is a simulation in a converging/diverging nozzle. What is it?

Imagine that we want to take a gas at rest and isentropically expand to supersonic speeds. The above equation shows that we must first accelerate the gas subsonically in a convergent duct. As soon as conditions are achieved, we must further expand the gas to supersonic speeds bby diverging the duct. Hence, a duct designed to achieve supersonic flow at its exit is a converging/diverging nozzle. This is the case of the simulation above. This also the reason why the the velocity from the (left to the right) along the nozzle increases although the middle of the nozzle narrows.

The opposite is also true, if we wish to take a supersonic flow and slow it down isentropically to subsonic speeds, we must first decelerate the gas in aconvergent duct, and the as soon as sonic flow is obtained, we must further decelerate it to subsonic speeds in a divergent duct. Hence a duct designed to achieve the subsonic flow is a converging/diverging diffuser.

Ref.:
(1) J. D. Anderson Jr., Fundamentals of Aerodynamics, Third Edition, New York: The Mc-Grawhill? Companies Inc., 2001.

Compressible Flow

By definition, compressibility of a fluid is

τ = (1/v)(dv/dp)

 

Where:

τ  = compressibility

v = specific volume (volume per unit mass)

p = pressure

The role of the compressibility in determining the properties of a fluid in motion is seen as follows. Define v as the specific volume (volume per unit mass). Hence, v = 1/ρ. Substituting this definition to the earlier equation, we obtain

τ = (1/ρ)(dρ/dp)

Thus, whenever the fluid experiences a change in pressure dp, the corresponding change in density from thi equation is

dρ = ρ x τ x dp.

Consider a fluid flow, if the fluid is a liquid where the compressibility is very small, the for a given pressure change dp from one point to another in the flow, this equation states that dρ will be negligibly small. In turn, we can reasonably assume that ρ is constant and that the flow of a liquid is incompressible. On the other hand, if the fluid is a gas, where the compressibility is large, then for a given pressure change dp from one point to another in flow, this equation states that dρ can be large. Thus, ρ is not constant, and in general, the flow of a gas is a compressible flow.

The exception to this is the low-speed flow of a gas; in such flows, the actual magnitude of the pressure changes (dp) throughout the flow field is small compared with the pressure itself. Thus, for a low-speed flow, dp in the equation above is small, and even though τ is large, the value of dρ can be dominated by the small dp. In such cases, ρ can be assumed to be constant, hence allowing us to analyze low-speed gas flows as incompressible flows.

Mach number is defined as:

M = V/a

Where:

M = Mach number

V = local flow velocity

a = local speed of a sound

It has been shown that when M larger than 0,3, the flow should be considered as compressible.

Ref.: 

 J. D. Anderson Jr., Fundamentals of  Aerodynamics, Third Edition, New York: The Mc-Grawhill Companies Inc., 2001.