Unit 3: Expansion waves and flow over nozzles - Subjective Questions

Prandtl–Meyer expansion: A Prandtl–Meyer expansion is a continuous, isentropic turning process in which a supersonic flow turns around a convex corner and accelerates through a fan of infinitesimal Mach waves.

Conditions for formation:

• The upstream flow must be supersonic, .
• The wall must turn away from the flow, creating a convex corner.
• The flow is generally assumed to be steady, two-dimensional, inviscid, and adiabatic.
• The expansion occurs through a continuous fan rather than through a single discontinuity.

Across the expansion fan:

• Mach number increases: .
• Static pressure, temperature, and density decrease.
• Stagnation properties remain constant because the process is isentropic.

For a calorically perfect gas, the Prandtl–Meyer function is

Here:

• is the angle through which a sonic flow must expand to attain Mach number .
• is the local Mach number.
• is the ratio of specific heats.
• is related to the geometry of supersonic Mach waves.

The reference value is . Since increases monotonically with , it can be used to determine the downstream Mach number after a specified expansion.

For an infinitesimal isentropic turning of a supersonic flow, the velocity change normal to a Mach wave gives

For adiabatic flow, the stagnation enthalpy is constant:

Using , , and , the velocity differential can be related to the Mach-number differential as

Substituting this result into the turning relation gives

Integrating from , where , to an arbitrary produces the Prandtl–Meyer function. For a finite expansion from to ,

Let the upstream Mach number be and the flow turn through an expansion angle .

Procedure:

1. Calculate the upstream Prandtl–Meyer angle .
2. Apply the turning relation:
   
3. Use a Prandtl–Meyer table, chart, or numerical inversion of to obtain corresponding to .
4. Verify that , as expected for an expansion.

Once is known, static-property ratios can be calculated using isentropic relations. The method is valid while the required value of does not exceed the limiting value .

A Prandtl–Meyer fan consists of an infinite number of weak expansion waves. Across the complete fan:

• The flow turns smoothly away from the wall.
• Velocity and Mach number increase.
• Static pressure, temperature, density, and speed of sound decrease.
• The Mach angle decreases as Mach number increases.
• Stagnation pressure and stagnation temperature remain constant.
• Entropy remains constant under the ideal inviscid assumption.

Thus, the expansion is an isentropic acceleration process, unlike a shock wave, which is irreversible and produces an entropy increase.

Because a Prandtl–Meyer expansion is isentropic, the stagnation properties are constant. For states 1 and 2,

Using the isentropic pressure relation,

Similarly, the density ratio is

Since , all three ratios are less than unity.

Prandtl–Meyer expansion:

• Forms when supersonic flow turns around a convex corner.
• Consists of a continuous fan of expansion Mach waves.
• Mach number and velocity increase.
• Pressure, temperature, and density decrease.
• The ideal process is isentropic, so stagnation pressure remains constant.

Oblique shock:

• Forms when supersonic flow is turned toward itself by a concave corner.
• Produces an abrupt discontinuity.
• Mach number and velocity generally decrease.
• Pressure, temperature, and density increase.
• Entropy increases and stagnation pressure decreases.

Both phenomena turn supersonic flow, but an expansion is smooth and reversible in the ideal model, whereas a shock is compressive and irreversible.

The maximum Prandtl–Meyer angle is obtained in the limiting case . Starting with

as , both inverse tangent terms approach . Therefore,

In degrees,

For air with ,

This is the theoretical angle required to expand a sonic flow to infinite Mach number.

The Prandtl–Meyer angle already acquired by the incoming flow is . The largest possible final value is , corresponding theoretically to .

Therefore, the maximum additional turning angle is

Interpretation:

• A sonic upstream flow has the greatest possible expansion range.
• As increases, increases and the remaining allowable turning decreases.
• If a proposed turn exceeds , no attached isentropic Prandtl–Meyer solution exists within the perfect-gas model.

A nozzle is ideally expanded when its exit static pressure equals the surrounding back pressure:

Its important characteristics are:

• No pressure adjustment is required outside the nozzle.
• The exhaust jet leaves without external expansion fans or compression shocks.
• For the specified stagnation conditions and area ratio, axial thrust is maximized without pressure mismatch losses.
• The pressure-thrust term becomes zero.

The thrust is then primarily the momentum thrust:

Ideal expansion occurs only at the nozzle's design pressure ratio.

A nozzle is under-expanded when its exit pressure is greater than the ambient back pressure:

The gas has not expanded sufficiently inside the nozzle. After leaving the exit:

• Expansion fans originate near the nozzle lip.
• The jet expands outward and its pressure decreases.
• Expansion waves may reflect from the jet boundary as compression waves.
• Repeated expansions and compressions can create a shock-cell pattern.
• At sufficiently large pressure ratios, the pattern may contain a Mach disk.

The external wave system adjusts the jet pressure toward . Under-expansion is common when a nozzle operates at a back pressure lower than its design value.

A nozzle is over-expanded when the nozzle exit pressure is lower than the ambient back pressure:

The pressure must increase to match the surroundings. This adjustment occurs through compression waves or shocks.

Possible behavior:

• For mild over-expansion, oblique shocks form near or outside the nozzle exit.
• With stronger over-expansion, a shock may stand inside the divergent section.
• The adverse pressure rise across the shock can cause boundary-layer separation.
• Separation may be asymmetric, producing side loads, vibration, and loss of thrust.
• A normal shock can reduce the flow from supersonic to subsonic.

Over-expansion occurs when the operating back pressure exceeds the nozzle's design exit pressure.

For a nozzle of fixed geometry and stagnation pressure:

• Lowering below the design value produces under-expansion.
• Setting equal to the design exit pressure gives ideal expansion.
• Raising above the design exit pressure produces over-expansion.

For one-dimensional isentropic flow, the area–Mach-number relation is

For a convergent–divergent nozzle operating on the supersonic branch, the exit area ratio determines the design exit Mach number . The corresponding exit pressure is

Thus:

• A larger generally produces a larger supersonic exit Mach number.
• A larger exit Mach number gives a lower design exit pressure.
• Comparing this calculated with identifies whether the nozzle is ideally expanded, under-expanded, or over-expanded.

As back pressure is progressively reduced:

1. At high , the flow is subsonic throughout the nozzle.
2. At the critical pressure ratio, the throat becomes choked and at .
3. A further reduction can produce supersonic flow followed by a normal shock in the divergent section.
4. As decreases, the normal shock moves downstream and the supersonic region grows.
5. The shock reaches the nozzle exit at a particular back pressure.
6. At lower , the nozzle flow is supersonic to the exit but may be over-expanded.
7. At the design back pressure, and the flow is ideally expanded.
8. Below the design back pressure, the jet is under-expanded and expands externally.

This sequence assumes suitable nozzle geometry and sufficiently high stagnation pressure.

A simple region is a supersonic-flow region in which one family of characteristic lines is straight and one corresponding Riemann invariant remains constant throughout the region.

For steady, two-dimensional, irrotational flow, the characteristic relations may be written as

with characteristic directions

A centered Prandtl–Meyer expansion fan is a classic simple-wave region. Its characteristics of one family are straight rays from the corner, and the local flow properties can be obtained from the invariant relations.

A non-simple region is a region where both families of characteristics influence the flow and neither Riemann invariant is constant throughout the entire region.

Differences:

• In a simple region, one characteristic family is straight and one invariant is uniform across the region.
• In a non-simple region, characteristics of both families are generally curved or spatially varying.
• Flow properties in a simple region can often be calculated directly from one compatibility relation.
• A non-simple region normally requires a characteristic mesh and simultaneous use of both compatibility equations.
• Non-simple regions commonly arise where waves interact, reflect, or are influenced by multiple boundaries.

Therefore, non-simple flow has a two-directional dependence, whereas simple-wave flow has a reduced one-family structure.

In a centered expansion fan, all expansion-wave characteristics of one family originate from the same convex corner. These rays are straight because the flow state is constant along each ray.

Using a consistent characteristic convention, one Riemann invariant remains constant throughout the fan, while the other changes from ray to ray. Consequently, the flow angle and Prandtl–Meyer angle satisfy a relation such as

or the corresponding relation with the opposite sign, depending on the orientation of the fan.

The local Mach number is obtained from , and the local characteristic direction is . Because only one independent wave family produces the variation, the centered fan is a simple region.

For steady, two-dimensional, inviscid, irrotational, isentropic supersonic flow, the governing equations can be combined along the characteristic directions. These directions make angles and with the reference axis, where

The differential compatibility relations are

After integration,

where and are constants on their respective characteristics. The characteristic slopes are

These equations form the basis of the method of characteristics for supersonic nozzle and wave-interaction problems.

Suppose a characteristic from a known point intersects a characteristic from a known point at point . With the convention

the compatibility equations at are

Adding and subtracting gives

The Mach number is found by inverting , after which and the thermodynamic properties are obtained from isentropic relations. Repeating this procedure creates the characteristic mesh for the non-simple region.