Unit 5: Brief introduction to the methods of characteristics - Subjective Questions

The method of characteristics (MOC) converts the governing hyperbolic partial differential equations of supersonic flow into ordinary differential equations along special lines called characteristics.

For a two-dimensional, steady, irrotational and isentropic supersonic flow, the characteristic directions are

where is the flow-deflection angle and is the Mach angle.

Significance:

• It determines the velocity, Mach number and flow direction throughout a supersonic flow field.
• It describes the propagation and interaction of compression and expansion waves.
• It is widely used to design shock-free supersonic nozzles.
• It replaces a difficult field problem with calculations along characteristic lines.
• It can treat regions in which linearized supersonic theory is inadequate.

For steady, two-dimensional, irrotational and isentropic flow, the governing equations become hyperbolic when . The two characteristic directions are

where

Introducing the Prandtl–Meyer function,

the compatibility relations are

• Along a characteristic: so that
• Along a characteristic: so that

At the intersection of a line carrying and a line carrying ,

The Mach number is then obtained by inverting the Prandtl–Meyer function.

A minimum-length nozzle expands a uniform sonic flow at the throat to a specified uniform supersonic exit flow in the shortest possible distance.

Procedure:

1. Determine the required exit Mach number and calculate .
2. For a sharp-cornered minimum-length nozzle, choose the maximum wall angle as
   
3. Divide the initial expansion fan into a number of equal angular increments.
4. Construct the and characteristic lines using
   
5. Use the compatibility relations and at each intersection.
6. Reflect characteristics appropriately at the centreline, where symmetry requires .
7. Form the nozzle wall from characteristic points so that the expansion waves are cancelled and the final wall angle becomes zero.

The exit flow is ideally parallel and uniform, with . Increasing the number of characteristics improves the accuracy of the computed contour.

For steady, two-dimensional, inviscid, small-disturbance subsonic flow, let the perturbation potential be . The linearized equation is

Define

and introduce the transformed coordinate

The potential equation then reduces to the incompressible Laplace equation in the transformed plane. Relating the compressible and incompressible perturbation velocities gives the Prandtl–Glauert pressure correction:

where is the corresponding incompressible pressure coefficient.

Consequently, for a thin two-dimensional airfoil,

The rule predicts an increase in pressure, lift and pitching-moment coefficients with Mach number. It becomes singular at , indicating failure of linear theory near the transonic regime.

The Prandtl–Glauert rule is

Assumptions:

• The flow is steady, inviscid and irrotational.
• The free-stream flow is subsonic.
• Disturbance velocities and airfoil slopes are small.
• The flow remains approximately isentropic.
• The airfoil is thin and normally operates at a small angle of attack.

Limitations:

• It becomes singular as .
• It cannot accurately represent local supersonic pockets or shock waves.
• It neglects nonlinear compressibility effects.
• It does not predict shock-induced separation or viscous drag.
• Accuracy decreases near the critical Mach number and for thick or highly cambered airfoils.

Thus, it is most reliable at low-to-moderate subsonic Mach numbers before significant transonic effects appear.

The Göthert rule extends linear subsonic compressibility transformation ideas to three-dimensional bodies. For

the transverse coordinates are transformed according to

while the streamwise coordinate remains unchanged. The compressible flow around the original body is related to incompressible flow around a body compressed in its transverse dimensions. The pressure relation is commonly expressed as

where is evaluated on the transformed incompressible body.

Difference from the Prandtl–Glauert rule:

• Prandtl–Glauert is most directly applied to thin two-dimensional airfoils and gives .
• Göthert explicitly transforms the transverse geometry and is useful for three-dimensional wings and bodies.
• For a two-dimensional thin airfoil, the geometry-scaling effect in the Göthert transformation reduces the result to the Prandtl–Glauert correction.
• Both are linearized rules and become unreliable near sonic conditions.

For a steady, two-dimensional, thin airfoil in a supersonic stream, the linearized small-disturbance equation is

Let

The general solution consists of disturbances propagating along Mach lines. Applying the tangency boundary condition at a surface having a small signed deflection angle gives the perturbation velocity and pressure relation. Since

Ackeret's formula becomes

for a compression surface. For an expansion surface,

Implications:

• Pressure is directly proportional to the local surface turning angle.
• Compression produces positive pressure coefficients.
• Expansion produces negative pressure coefficients.
• The formula is valid for thin surfaces, small angles and attached supersonic flow.
• It does not accurately represent strong shocks or large flow deflections.

Consider a flat plate at a small angle of attack in a supersonic stream. Define

Ackeret's theory gives

where the subscripts and denote the lower and upper surfaces. Therefore, the pressure difference is

For small angles, the normal-force coefficient is approximately the lift coefficient:

The pressure force is nearly normal to the plate. Its streamwise component gives the wave drag:

Here must be expressed in radians. The result shows that inviscid supersonic flow produces wave drag even for a zero-thickness plate when it operates at a nonzero angle of attack.

Let a symmetric double-wedge airfoil have a small semi-wedge angle and operate at a small angle of attack . According to Ackeret's theory, the pressure coefficient on each panel is proportional to its local flow-turning angle:

After resolving and summing the panel pressures, the lift coefficient is

The wave-drag coefficient is

Interpretation:

• The term is lift-dependent wave drag.
• The term is thickness-dependent wave drag.
• At zero angle of attack, the airfoil has zero lift but finite wave drag:
  
• Increasing airfoil thickness or angle of attack increases wave drag.

The result assumes small angles, a thin airfoil, attached waves and uniform supersonic free-stream flow.

Let the total velocity potential be

where is a small perturbation potential. Thus,

Linearizing the continuity, momentum and isentropic relations gives the two-dimensional small-disturbance equation

Its mathematical character depends on the free-stream Mach number:

• Subsonic flow, : , so the equation is elliptic.
• Sonic flow, : the coefficient of vanishes, and the linear equation is degenerate.
• Supersonic flow, : , so the equation is hyperbolic.

In transonic flow, local Mach numbers cross unity, so the governing nonlinear equation is of mixed elliptic–hyperbolic type. This change of type explains the fundamental differences in disturbance propagation among the flow regimes.

The linearized two-dimensional small-perturbation equation for subsonic flow is

with . Since both second-derivative coefficients have the same sign, the equation is elliptic.

Using

the equation can be transformed into Laplace's equation.

Physical characteristics:

• Disturbances can influence points both upstream and downstream.
• There are no real characteristic lines representing a finite domain of dependence.
• Boundary conditions must generally be specified around the complete body and at infinity.
• Compressibility amplifies pressure disturbances relative to incompressible flow.
• Linear theory is effective only while perturbation velocities remain small and no strong local transonic effects occur.

Near Mach one, the linear coefficient becomes small. Terms that are formally nonlinear can therefore be as important as the linear terms and cannot be neglected. A common form of the two-dimensional transonic small-disturbance equation is

Important features:

• The coefficient of depends on the local perturbation velocity.
• The equation is elliptic where the local flow is subsonic.
• It is hyperbolic where the local flow is supersonic.
• A transonic flow field can therefore contain both equation types.
• Shock waves may terminate local supersonic regions.

The Prandtl–Glauert rule fails because it predicts unbounded pressure as . The nonlinear transonic equation instead accounts approximately for local acceleration through sonic speed and for weak-shock formation.

For , the linearized small-perturbation equation can be written as

Define

The equation is hyperbolic and may be factored as

Its characteristic lines satisfy

where

Physical meaning:

• Disturbances propagate along Mach lines.
• A point is influenced only by conditions within its upstream Mach cone.
• Small disturbances cannot propagate upstream against a uniform supersonic stream.
• Initial-value and characteristic methods are therefore suitable for supersonic-flow calculations.

Hypersonic flow generally refers to flow at approximately . Although a body may be slender, ordinary linear theory can fail because the product of Mach number and body slope need not be small.

For a slender body with characteristic slope or thickness ratio , an important similarity parameter is

Interpretation:

• If , linearized supersonic theory may remain useful.
• If , nonlinear hypersonic small-disturbance effects must be retained.
• Strong compression, high temperatures and large density changes may occur behind the shock.
• Shock waves lie close to slender bodies, producing a thin shock layer.
• At very high speeds, real-gas effects such as vibrational excitation, dissociation and ionization may become important.

For sufficiently large hypersonic Mach numbers, Newtonian impact theory gives the approximate surface pressure relation

where is the local surface inclination. Thus, hypersonic theory generally requires nonlinear and thermochemical considerations beyond ordinary linear small-perturbation theory.

Subsonic flow:

• Equation:
• Type: elliptic.
• Disturbances can propagate upstream and downstream.

Transonic flow:

• A nonlinear coefficient must be retained:
  
• Type: mixed elliptic–hyperbolic.
• Local supersonic pockets and shock waves can occur.

Supersonic flow:

• Equation:
• Type: hyperbolic.
• Disturbances are confined by Mach lines and cannot travel upstream.

Hypersonic flow:

• The equation is fundamentally hyperbolic, but nonlinear terms may remain important even for slender bodies.
• The similarity parameter controls the applicability of linear theory.
• Strong shocks, thin shock layers and real-gas effects can arise.

Thus, increasing Mach number changes both the mathematical nature of the governing equations and the physical propagation of disturbances.

A typical experiment uses a low-speed wind tunnel with an airfoil model mounted in the test section.

Main equipment:

• A model with pressure taps distributed over the upper and lower surfaces.
• A multi-tube manometer or electronic pressure scanner.
• A force balance for measuring lift, drag and pitching moment.
• A Pitot-static tube for free-stream dynamic pressure.
• An angle-of-attack adjustment mechanism.

Procedure:

1. Establish a uniform free-stream velocity .
2. Record atmospheric conditions and calculate density and Reynolds number.
3. Set the required angle of attack .
4. Measure surface pressures and balance forces after steady conditions are reached.
5. Repeat over a range of angles of attack.
6. Calculate , , and .

Corrections may be required for tunnel-wall interference, blockage, support forces, balance tare, flow nonuniformity and finite-span effects. The resulting plots usually include versus , versus and versus .

For small and moderate angles of attack, the lift coefficient varies approximately linearly with angle:

where is the lift-curve slope and is the zero-lift angle.

Observed behavior:

• A symmetric airfoil has approximately .
• A positively cambered airfoil normally has a negative zero-lift angle.
• Thin-airfoil theory predicts the two-dimensional slope
  
• Real measurements may show a lower slope because of viscosity, finite aspect ratio and tunnel effects.
• As increases, the adverse pressure gradient strengthens and boundary-layer separation grows.
• At the stall angle, the airfoil reaches .
• Beyond stall, lift generally decreases and drag rises rapidly.

The exact stall behavior depends on airfoil geometry, Reynolds number, surface roughness and free-stream turbulence.

The pressure coefficient at each surface tap is calculated from

The measured values are plotted against for the upper and lower surfaces. The pressure difference is

For a thin airfoil at a small angle, the normal-force coefficient is approximately

The pitching-moment coefficient about the leading edge is approximately

The moment about another reference position is obtained using

Numerical integration, such as the trapezoidal rule, is applied to discrete pressure-tap data. Pressure integration gives pressure forces and moments but does not directly include skin-friction drag.

Airfoil drag can be measured using either a calibrated force balance or a wake survey.

In the wake-survey method, the loss of streamwise momentum behind the airfoil is measured. For an approximately incompressible two-dimensional wake, drag per unit span is obtained from the momentum deficit:

with appropriate pressure corrections when static pressure has not recovered to the free-stream value. The drag coefficient is

A drag polar is a plot of against . It is often approximated over a limited range by

Significance:

• represents profile drag near zero lift.
• The curve identifies the minimum-drag condition.
• It permits estimation of aerodynamic efficiency .
• Sudden drag growth can indicate separation or stall.
• Comparing polars reveals the effects of Reynolds number, roughness and airfoil geometry.

The Reynolds number is

It controls the relative importance of inertial and viscous forces and strongly affects the airfoil boundary layer.

Effects of Reynolds number:

• At low , laminar separation may occur early and form a separation bubble.
• Increasing generally delays separation, increases and can raise the stall angle.
• Profile drag often decreases as Reynolds number increases over practical low-speed ranges.

Boundary-layer transition:

• A laminar boundary layer has lower skin-friction drag but separates more readily under an adverse pressure gradient.
• A turbulent boundary layer has greater skin friction but better resistance to separation.
• Transition location therefore affects drag, lift and stall behavior.

Surface roughness:

• Roughness can trigger premature transition.
• It generally increases drag and may reduce maximum lift.
• Contamination near the leading edge can cause earlier separation and abrupt stall.

Accurate experiments should therefore control model finish, free-stream turbulence and Reynolds number, and should document whether transition is natural or artificially fixed.