PATTS REVIEW CENTERPATTS COLLEGE OF AERONAUTICS

HIGH-SPEED AERODYNAMICS

REVIEWER: R.R. RENIGEN

COMPRESSIBLE FLOW

When a change in pressure is accompanied by such a change in density, the flow is called compressible and the amount of compressibility depends on the velocity of the air.

At supersonic speeds, all pressure changes are accompanied either by shock waves, through which the pressure and density are increased, or else by expansion waves, through which these quantities are reduced.

GASESA gas is composed of individual, distinct particles, each in continual, irregular motion, and these particles are

constantly colliding with each other.

1 Perfect Gas Law equation

P = RT

Where: P = Pressure in Pa = Density in Kg/m3

R = Universal gas constant ( = 287.08 J/Kg-K for air) T = Absolute temperature in K

2 Adiabatic Process

Where:

= = specific heat ratio ( = 1.4 for dry air )

CP = specific heat at constant pressure ( = 1,006 J/Kg-K )CV = specific heat at constant volume ( = 718.6 J/Kg-K )

3 Speed of Sound in Air, Va Speed of sound – sound waves travel through the air at a definite speed.

PVa =

1

Va =

Va = Va in m/s , T in K

4 Compressible Bernoulli Equation

constant

Where: constant

1 2

OR:

5 Stagnation Pressure in Compressible FluidsAt the stagnation point, the velocity is zero and the pressure and density are at maximum.

6 Mach Number

VM =

Va

7 Reynolds Number

VRRN =

where:

V = average axial velocityR = inner radius of tube = dynamic viscosity of air

8 Law of Continuity

AV = Constant where : constant1 2

2

1A1V1 = 2A2V2

Relevant Properties of gases

Types of gas Ratio of SpecificHeats

Gas ConstantR( J/Kg-K)

Air 1.4 287.08Carbon dioxide (CO2) 1.288 188.96Hydrogen (H2) 1.4 4125.52Helium (He) 1.666 2077.67Neon (Ne) 1.666 412.10Argon (A) 1.666 208.17Oxygen (O2) 1.395 259.90Nitrogen (N2) 1.399 296.86

One-dimensional Flow – Nozzles and Diffusers

One-dimensional flow is used to describe a flow in which flow conditions are sensibly constant across a cross-section of a nozzle or diffuser normal to the flow direction. Conditions change only along a stream line from one cross-section to the other along the length of the nozzle or diffuser. In two-dimensional flow, condition changes across a cross-section.

Thermodynamic Relations – Isentropic or Reversible Adiabatic Gas Laws apply in a flow where no compression shock waves occur. Note that isentropic losses in pressure exist through shock waves.

1. Pressure, Density, Temperature, Speed of Sound and Mach Number Equations

For any gas

2 P2 2 T2 -1 Va2 -1 (-1) M1

2 + 2 -1 = = = =

P1 1 T1 Va1 (-1) M22 + 2

For air, = 1.4

1. 4 3.5 7 3.5 P2 2 T2 Va2 M1

2 + 5 = = = = P1 1 T1 Va1 M2

2 + 5

2. Area – Mach Number Relationship

3

For any gas

For air 3

A2 M1 M22 + 5

= A1 M2 M1

2 + 5

Problems:

1. The temperature and pressure at the stagnation point of a high-speed missile are 934R and 7.8 atm, respectively. Calculate the density at this point.

3. Consider the isentropic flow over an airfoil. The freestream conditions are T= 245 K and P= 4.35 104N/m2. At a point on the airfoil, the pressure P = 3.6 104N/m2. Calculate the density at this point.

4. Consider the isentropic flow through a supersonic wind-tunnel nozzle. The reservoir properties are To= 500 K and Po = 10 atm. If P = 1 atm at the nozzle exit, calculate the exit temperature and density.

5. In the reservoir of a supersonic wind tunnel, the velocity is negligible, and the temperature is 1000 K. The temperature at the nozzle exit is 600 K. Assuming adiabatic flow through the nozzle, calculate the velocity at the exit.

6. An airfoil in a freestream where P = 0.61 atm, = 0.819 Kg/m3, and V = 300 m/s. At a point on the airfoil surface, the pressure is 0.5 atm. Assuming isentropic flow, calculate the velocity at that point.

7. An air tank with a nozzle has a pressure of 196.32 KPa and density of 1.9 Kg/m3. Outside the converging-diverging nozzle, the pressure is atmospheric and designed to have a Mach No. of 1.0 and 1.5 at the throat and exit respectively. The area at the throat is 0.11m2. Calculate the following: (a) Temperature and speed of sound at the tank. (b) Pressure, density, temperature and speed of sound at the throat. (c) Mass flow at the exit.

MACH NUMBERS AND SHOCK WAVES

Mach Number Classification

4

1. Incompressible , M 0.32. Subsonic , M 1.03. Sonic , M 1.04. Transonic , 0.8 M 1.25. Supersonic , 1.0 M 56. Hypersonic , M > 5

Shock wave

A large-amplitude compression wave, such as that produced by an explosion, caused by supersonic motion of a body in motion.

Manifest the collapsed area of the dense region where the density is instantaneous.Formed thru the continuous compilation of particles travelling at high speeds that form a boundary line.

Reaction from a Shock Wave

a. Compression of Gases

When the flow is supersonic, compression does not occur gradually, but takes place very suddenly in a thin region, which is known as a shock wave.

It is the region of increasing pressure and density with falling velocity.

Compressions are propagated as finite disturbances, at a speed greater than the speed of sound.

b. Expansion of Gases

Regions in which the velocity increases, while pressure and density decrease are known as expansion regions.

Disturbances in the flow which constitute expansions are propagated as infinitesimal disturbances with speed of sound.

Types of Waves

a. shock waves/compression waves

1. Normal Shock Waves (NSW) formed by blunt bodies

2. Oblique Shock Waves (OSW) a function of deflecting angles

b. expansion waves

When the density is decreased, the change is gradual rather than as in the compressive case, and is always of the oblique type. Because the change is gradual, it is not a shock wave and has no normal type corresponding to the compressive case.

5

Examples of Normal Shocks

1. Flow over a blunt body

The flow is supersonic over a blunt bodyA strong bow shock wave exists in front of the body.Although this wave is curved, the region of the shock closest to the nose is essentially normal to the flow.The streamline that passes through this normal portion of the bow shock later impinges on the nose of the body and controls the values of stagnation pressure and temperature at the nose.

2. Overexpanded flow through a nozzle

6

Po

This portion of the bow shock is normal to the flow

bow shock

V

M > 1

Normal shock inside the nozzle

M > 1M < 1

Supersonic flow is established inside a nozzle (which can be a supersonic wind tunnel, a rocket engine, etc.) where the back pressure is high enough to cause a normal shock wave to stand inside the nozzle.

Example of Oblique Shock Wave

The wall is turned upward at the corner through the deflection angle ; i.e., the corner is concave.The flow at the wall must be tangent to the wall; hence, the streamline at the wall is also deflected upward through the angle .The bulk of the gas is above the wall, the streamlines are turned upward, into the main bulk of the flow.Whenever a supersonic flow is “turned into itself”, an oblique shock wave will occur.

The originally horizontal streamlines ahead of the wave are uniformly deflected in crossing the wave, such that the streamlines behind the wave are parallel to each other and inclined upward at the deflection angle .Across the wave, the Mach number discontinuously decreases, and the pressure, density, and temperature discontinuously increase.

Example of Expansion wave

7

M1 > 1 P1

1

T1

M2 < M1

P2 > P1

2 > 1

T2 > T1

/ / / / / / / / / / / / / / / /

Oblique shock1 2

concave corner

/ / / / / / / / / / / / / / / /

Expansion fan

1

Shows the case where the wall is turned downward at the corner through the deflection angle ; i.e., the corner is convex.The flow at the wall must be tangent to the wall; hence, the streamline at the wall is deflected downward through the angle .The bulk of the gas is above the wall, the streamlines are turned downward, away from the main bulk of the flow.Whenever a supersonic flow is “turned away from itself”, an expansion wave will occur. This expansion wave is in the same of a fan centered at the corner. The fan continuously opens in the direction away from the corner.The originally horizontal streamlines ahead of the expansion wave are deflected smoothly and continuously through the expansion fan such that the streamlines behind the wave are parallel to each other and inclined downward at the deflection angle .Across the expansion wave, the Mach number increases, and the pressure, temperature, and density decrease.

Water-Wave Analogy Supersonic Speed

Disturbance pattern created by particles moving faster than speed of sound.

The speed of the particle is increase until it is greater than the speed at which the pressure waves travel (V>Va). In the case, the object travels faster than the wavelets it produces, and the individual waves combine along a common front, where the wavelets intersect, reinforce each other, and create a new and much stronger wave along the tangent common to all wavelets.

8

V > Va

5 4 3 2 1

Vat

Vt

M2 < M1

P2 > P 1

2 > 1

T2 > T1

M1 >1

P1

1

T1

2

The portion of the wavelets ahead of the point to tangency will lose their identity and be merge into the envelope created by the other wavelets.Replacing the wavelets in water by pressure impulses in the air, it can be seen that a definite line of demarcation is set up in the air, separating the region affected by the body from the free-stream conditions.

Mach Line, Angle and Number

Mach Line

The line of disturbance created along the envelope of individual wavelets.

The line so drawn to evaluate the geometric condition of a supersonic pattern (i.e., Mach cone).

The point of contact with the circles of their common tangent is the location of the source.

The disturbance at this point tends to build up into a much stronger disturbance than the one being created by the source; but since the latter is infinitesimal, the disturbance remains vanishingly weak.

There is still no change in flow properties across this common tangent which, however, divides the region which is affected by the disturbance from that which is not. This is known as a normal Mach line.

Mach Number, M

9

V2 - Va2

Va

V

Mach line

M > 1

/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

The ratio between the speed of the air and the speed of sound in the air.

Mach Angle,

The angle which the Mach line makes with the free-stream direction.

Defined by the relative velocities between the free airstream V, and the speed of sound in the stream Va.

Problems:

1. A supersonic aircraft flies horizontally at 3,000 meters altitude with a constant velocity of 800 meters per second. The aircraft passes directly overhead a stationary ground observer. How much time elapses after it has passed over the observer before the latter hears the noise from the aircraft?

2. A jet plane flies at an altitude of 2,000 meters. An observer on the ground notes that he hears the sound of the plane exactly 5 seconds after the plane has passed directly overhead. Calculate the velocity of the plane.

Normal Shock Wave Equations

1. Mach Number

For any gas For air, γ=1.4

10

Unknown conditions behind the wave

V2

P2 2 T2 Va2 M2

V1

P1 1 T1 Va1 M1

Given conditions ahead of the wave

Total head pressure across the NSW

=

2. Velocity Ratio

=

3. Density Ratio

=

4. Pressure Ratio

=

5. Temperature Ratio

=

6. Stagnation Pressure

= = = Ahead of the shock

= = = Behind the shock

= =

Problems:

11

1. Consider a normal shock wave in air where the upstream flow properties are V1 = 680 mps, T1 = 288 K and P1 = 1 atm. Calculate the velocity, temperature, and pressure downstream of the shock.

2. A normal shock wave was formed on the surface of a supersonic aircraft at a velocity of 1,600 m/s into still atmospheric air at standard seal level conditions. Calculate: a)M1 b)M2 c)P2 d)T2 e)V2.

3. The pressure upstream of a normal shock wave is 1 atm. The pressure and temperature downstream of the wave are 10.33 atm and 1390 °R, respectively. Calculate the Mach number and temperature upstream of the wave.

4. Air at initial velocity would cause a pressure ratio of 3 across a normal shock wave at 6 km above sea level. Find: a)V1 b)P2 c)ρ2 d)V2 e) f)T2.

5. At an altitude of 7 km, air is travelling at a supersonic speed and decelerated by a normal shock wave which causes a density ratio of 0.325 across the wave. Compute: a) M1 b)V 1 c) d)M2 e)V2 f)P2 g)ρ2 h)T2.

6. The flow just upstream of a normal shock wave is given by P1 = 1 atm, T1 = 288 K, M1 = 2.6. Calculate the following properties just downstream of the shock: P2, T2, M2, and .

7. Air at 10 km above sea level, initially travelling at supersonic speed is decelerated thru a normal shock wave. The deceleration caused a velocity of 1,600 Kph after the wave. Calculate the following:

a) Velocity and Mach number before the wave

b) P, ρ, T, and Va after the wave.

Oblique Shock Wave Equation (Exact Method): Two-dimensional flow

Where: θW = wave angle , θ = deflection angle

The effect of viscosity in the air and in the formation of a boundary layer is neglected. It is assumed that the flow over the surface is not affected by frictional forces. Separation and wake effects are likewise neglected.

12

The flow is assumed to be irrotational. This will be true only when the air deflected along a surface. Whenever the surface in the airstream has curvature, the initial wave will be curved and the flow will have vorticity. The curvature of the wave is small if the curvature of the surface is and the effect of vorticity will be correspondingly small.

No heat conduction exists between the adjacent streamline or across the shock wave. The error incurred by this assumption is negligible.

As with the approximate theory, the shock wave must be attached to the corner or leading edge of the surface over which the air is flowing.

Ahead of the shock

,

Or:

,

After the shock

,

,

1. Pressure Ratio For any gas For air, γ=1.4

=

Or:

=

13

2. Density Ratio

=

Or:

3. Temperature Ratio

=

4. Normal Component of M 2

=

5. Mach number after the wave

Or:

=

Or:

=

6. Wave Angle2

14

=

7. Deflection Angle

8. Velocity Ratio

9. Total Upstream Pressure

= = =

10. Total Downstream Pressure

= = =

Problems:

1. Using the exact theory method, find the final Mach number and density in the compressive case of an initial Mach number of 1.75 and a deflection of 8°. Assume standard sea level conditions.

2. Find all aerodynamic and thermodynamic conditions on both sides of compression wave. Supersonic stream is at M=1.6 compressing through an angle of 8°. Initial pressure and temperature are 10 psia and 20°F, respectively.

3. Consider a supersonic flow with M=2, P=1 atm, and T=288 K. This flow is deflected at a compression corner through 20°. Calculate M, P, T, and P0 and T0 behind the resulting oblique shock wave.

4. Consider an oblique shock wave with a wave angle of 30°. This upstream flow Mach number is 2.4. Calculate the deflection angle of the flow, the pressure and temperature ratios across the shock wave and the Mach number behind the wave.

5. Consider an oblique shock wave with θW = 35° and a pressure ratio . Calculate the upstream Mach

number.

15

2 24° 25° 26° 27° 28° 29° 30° 3°1 32° 33° 34° 35° 36° 37° 38° 39° 40° 41° 42° 43° 44° 45° 46° 47° 48° 49° 50° 51° 52° 53° 54° 55° 56° 57° 58° 59° 60° 61° 62° 63° 64° 65° 66°

1° 2.530 2.433 2.344 2.261 2.185 2.115 2.050 1.989 1.932 1.879 1.830 1.783 1.740 1.699 1.660 1.624

1.589

1.557

1.526

1.498

1.470

1.444

1.420

1.396

1.374

1.353

1.333

1.315

1.297

1.280

1.264

1.248

1.234

1.220

1.207

1.194

1.183

1.172

1.161

1.152

1.142

1.134

1.126

2° 2.608 2.505 2.411 2.324 2.244 2.171 2.102 2.039 1.980 1.924 1.873 1.825 1.779 1.737 1.697 1.660

1.624

1.591

1.560

1.530

1.502

1.475

1.450

1.426

1.404

1.382

1.362

1.343

1.325

1.308

1.292

1.276

1.262

1.248

1.235

1.223

1.211

1.201

1.190

1.181

1.172

1.164

1.157

3° 2.691 2.582 2.483 2.391 2.307 2.230 2.158 2.091 2.030 1.972 1.918 1.868 1.821 1.777 1.736 1.697

1.661

1.626

1.594

1.563

1.534

1.507

1.481

1.457

1.434

1.412

1.392

1.372

1.354

1.337

1.320

1.305

1.290

1.276

1.264

1.251

1.240

1.230

1.220

1.211

1.203

1.195

1.188

4° 2.782 2.666 2.560 2.463 2.374 2.293 2.217 2.147 2.083 2.022 1.966 1.914 1.865 1.819 1.776 1.736

1.698

1.663

1.629

1.598

1.568

1.540

1.514

1.489

1.465

1.443

1.422

1.402

1.384

1.366

1.349

1.334

1.319

1.306

1.293

1.281

1.270

1.259

1.250

1.241

1.233

1.226

1.220

5° 2.881 2.757 2.644 2.541 2.447 2.360 2.280 2.207 2.139 2.076 2.017 1.962 1.911 1.864 1.819 1.777

1.738

1.701

1.666

1.634

1.603

1.574

1.547

1.522

1.497

1.475

1.453

1.433

1.414

1.396

1.379

1.364

1.349

1.335

1.322

1.311

1.300

1.290

1.280

1.272

1.265

1.258

1.252

6° 2.991 2.857 2.735 2.625 2.525 2.433 2.348 2.270 2.199 2.132 2.071 2.013 1.960 1.910 1.863 1.820

1.779

1.741

1.705

1.671

1.640

1.610

1.582

1.555

1.531

1.507

1.485

1.465

1.445

1.427

1.410

1.394

1.379

1.366

1.353

1.341

1.330

1.320

1.311

1.303

1.296

1.290

1.285

7° 3.112 2.967 2.836 2.717 2.609 2.511 2.421 2.339 2.263 2.193 2.128 2.067 2.011 1.959 1.910 1.865

1.822

1.783

1.745

1.710

1.678

1.647

1.618

1.591

1.565

1.541

1.519

1.497

1.478

1.459

1.442

1.426

1.411

1.397

1.384

1.372

1.361

1.352

1.343

1.335

1.329

1.323

1.319

8° 3.248 3.089 2.946 2.818 2.702 2.597 2.500 2.412 2.332 2.257 2.188 2.125 2.066 2.011 1.960 1.912

1.868

1.826

1.788

1.751

1.717

1.685

1.655

1.627

1.601

1.576

1.553

1.531

1.511

1.492

1.474

1.458

1.443

1.429

1.416

1.404

1.393

1.384

1.375

1.368

1.362

1.356

1.353

9° 3.402 3.226 3.070 2.930 2.804 2.690 2.587 2.492 2.406 2.327 2.254 2.186 2.124 2.066 2.013 1.963

1.916

1.873

1.832

1.794

1.759

1.725

1.694

1.665

1.638

1.612

1.589

1.566

1.545

1.526

1.508

1.491

1.476

1.462

1.449

1.437

1.426

1.417

1.408

1.401

1.395

1.391

1.387

10° 3.579 3.382 3.208 3.054 2.917 2.793 2.681 2.579 2.487 2.402 2.324 2.252 2.186 2.125 2.068 2.016

1.967

1.921

1.879

1.839

1.802

1.767

1.735

1.705

1.677

1.650

1.626

1.603

1.581

1.561

1.543

1.526

1.510

1.496

1.482

1.470

1.460

1.450

1.442

1.435

1.430

1.425

1.423

11° 3.784 3.561 3.366 3.195 3.043 2.907 2.785 2.675 2.575 2.484 2.400 2.324 2.253 2.188 2.128 2.072

2.021

1.973

1.928

1.887

1.848

1.812

1.778

1.746

1.717

1.690

1.664

1.640

1.618

1.598

1.579

1.561

1.545

1.530

1.517

1.505

1.494

1.485

1.477

1.470

1.465

1.461

1.459

12° 4.026 3.769 3.548 3.355 3.185 3.035 2.901 2.780 2.671 2.573 2.483 2.401 2.325 2.256 2.192 2.133

2.078

2.027

1.980

1.937

1.896

1.858

1.823

1.790

1.759

1.731

1.704

1.680

1.657

1.636

1.616

1.598

1.582

1.567

1.553

1.541

1.530

1.521

1.513

1.506

1.501

1.498

1.496

13° 4.320 4.017 3.760 3.540 3.348 3.180 3.031 2.898 2.779 2.671 2.573 2.485 2.403 2.329 2.261 2.198

2.140

2.086

2.036

1.990

1.947

1.907

1.870

1.836

1.804

1.774

1.746

1.721

1.697

1.675

1.655

1.636

1.619

1.604

1.590

1.578

1.567

1.558

1.550

1.544

1.539

1.536

1.534

14° 4.686 4.318 4.014 3.757 3.537 3.346 3.178 3.030 2.898 2.780 2.673 2.577 2.489 2.409 2.335 2.268

2.206

2.148

2.096

2.047

2.001

1.959

1.920

1.884

1.850

1.819

1.790

1.764

1.739

1.716

1.695

1.676

1.659

1.643

1.629

1.616

1.605

1.596

1.588

1.582

1.577

1.575

1.574

15° 5.162 4.697 4.325 4.018 3.759 3.539 3.348 3.181 3.033 2.902 2.784 2.678 2.582 2.495 2.416 2.343

2.277

2.216

2.159

2.107

2.059

2.015

1.973

1.935

1.900

1.867

1.837

1.809

1.783

1.759

1.738

1.718

1.700

1.683

1.669

1.656

1.645

1.635

1.628

1.622

1.617

1.615

1.614

16° 5.815 5.193 4.717 4.338 4.027 3.767 3.545 3.354 3.187 3.039 2.908 2.791 2.686 2.591 2.505 2.426

2.354

2.288

2.228

2.172

2.121

2.074

2.030

1.989

1.952

1.918

1.886

1.856

1.829

1.805

1.782

1.761

1.743

1.726

1.711

1.697

1.686

1.676

1.669

1.663

1.659

1.656

1.656

17° 6.795 5.885 5.238 4.747 4.360 4.044 3.781 3.557 3.365 3.197 3.049 2.919 2.802 2.697 2.602 2.516

2.438

2.367

2.302

2.242

2.187

2.137

2.090

2.047

2.008

1.971

1.938

1.907

1.878

1.852

1.828

1.807

1.787

1.770

1.754

1.741

1.729

1.719

1.711

1.705

1.701

1.700

1.700

18° 8.519 6.948 5.975 5.296 4.788 4.390 4.067 3.800 3.574 3.380 3.211 3.063 2.932 2.815 2.711 2.616

2.531

2.453

2.383

2.318

2.259

2.205

2.155

2.109

2.067

2.028

1.993

1.960

1.930

1.902

1.877

1.855

1.834

1.816

1.800

1.786

1.774

1.764

1.756

1.750

1.746

1.744

1.745

19° 12.993

8.913 7.139 6.086 5.368 4.839 4.428 4.098 3.825 3.596 3.400 3.230 3.081 2.949 2.832 2.728

2.634

2.548

2.471

2.401

2.337

2.278

2.225

2.176

2.131

2.089

2.051

2.016

1.985

1.956

1.929

1.905

1.884

1.865

1.848

1.833

1.821

1.810

1.802

1.796

1.792

1.791

1.792

20° 14.922

9.424 7.376 6.222 5.457 4.902 4.476 4.136 3.857 3.623 3.424 3.253 3.102 2.970 2.853

2.748

2.654

2.569

2.492

2.422

2.358

2.300

2.247

2.199

2.154

2.114

2.077

2.043

2.012

1.984

1.959

1.936

1.916

1.898

1.883

1.870

1.859

1.851

1.845

1.841

1.840

1.842

21° 18.470

10.103

7.671 6.388 5.564 4.978 4.533 4.182 3.896 3.657 3.454 3.280 3.128 2.995

2.877

2.772

2.677

2.592

2.516

2.446

2.383

2.325

2.273

2.225

2.181

2.142

2.105

2.072

2.043

2.016

1.992

1.970

1.952

1.935

1.922

1.911

1.902

1.896

1.892

1.891

1.893

22° 28.325

11.043

8.040 6.590 5.693 5.069 4.602 4.237 3.941 3.697 3.490 3.313 3.159

3.024

2.905

2.799

2.704

2.619

2.543

2.473

2.410

2.353

2.301

2.254

2.211

2.172

2.137

2.105

2.076

2.051

2.028

2.008

1.991

1.977

1.965

1.956

1.949

1.946

1.945

1.947

23° 12.430

8.512 6.836 5.847 5.176 4.683 4.302 3.995 3.743 3.531 3.351

3.194

3.057

2.937

2.830

2.735

2.650

2.573

2.504

2.441

2.385

2.333

2.287

2.244

2.206

2.172

2.141

2.114

2.089

2.068

2.050

2.035

2.022

2.013

2.006

2.002

2.002

2.004

24° 14.722

9.131 7.138 6.031 5.303 4.778 4.377 4.058 3.797 3.579

3.394

3.235

3.096

2.974

2.866

2.770

2.684

2.607

2.538

2.476

2.419

2.369

2.323

2.281

2.244

2.211

2.181

2.155

2.132

2.113

2.096

2.083

2.073

2.066

2.062

2.062

2.065

25° 19.507

9.974 7.515 6.253 5.454 4.890 4.465 4.131 3.860

3.635

3.444

3.281

3.140

3.016

2.906

2.810

2.723

2.646

2.577

2.514

2.458

2.408

2.363

2.322

2.286

2.254

2.226

2.201

2.180

2.163

2.148

2.137

2.130

2.126

2.125

2.129

26° 43.965

11.198

7.995 6.523 5.633 5.021 4.567 4.215

3.932

3.698

3.502

3.334

3.189

3.063

2.952

2.854

2.767

2.689

2.620

2.558

2.502

2.452

2.408

2.368

2.333

2.302

2.275

2.253

2.233

2.218

2.206

2.198

2.194

2.193

2.197

27° 13.168 8.625 6.855 5.846 5.174 4.686

4.312

4.014

3.770

3.567

3.394

3.245

3.116

3.003

2.904

2.816

2.737

2.668

2.606

2.550

2.501

2.457

2.419

2.385

2.356

2.331

2.310

2.293

2.280

2.271

2.266

2.266

2.269

28° 17.075

9.492 7.271 6.103 5.355

4.825

4.424

4.109

3.853

3.641

3.462

3.309

3.176

3.061

2.960

2.870

2.792

2.722

2.660

2.605

2.556

2.513

2.476

2.443

2.416

2.393

2.374

2.360

2.350

2.344

2.344

2.348

29° 32.162

10.770

7.809 6.418

5.571

4.987

4.554

4.218

3.948

3.726

3.539

3.381

3.244

3.126

3.023

2.932

2.852

2.782

2.720

2.665

2.617

2.575

2.539

2.509

2.483

2.462

2.446

2.435

2.429

2.428

2.432

30° 12.897

8.530

6.811

5.831

5.179

4.706

4.344

4.057

3.822

3.627

3.462

3.321

3.199

3.094

3.001

2.920

2.849

2.787

2.733

2.686

2.645

2.611

2.582

2.559

2.541

2.528

2.521

2.519

2.523

FUNCTIONS FOR CALCULATIONS OF COMPRESSIVE FLOW

16

Prandtl-Mayer Expansion Waves

The expansion fan is a continuous expansion region which can be visualized as an infinite number of Mach

waves, each making the Mach angle μ = sin-1 with the local flow direction.

The expansion fan is bounded downstream by a Mach wave which makes the angle μ1, with respect to the

upstream flow, where μ1 = sin-1 . The expansion fan is bounded downstream by another Mach wave which

makes the angle μ2 with respect to the downstream flow, where μ2 = sin-1 . Since the expansion through the

wave takes place across a continuous succession of Mach waves, and since ds = 0 for each Mach wave, the expansion is isentropic. This is in direct contrast to flow across an oblique shock, which always experience an entropy increase.

An expansion wave emanation from a sharp convex corner is called a centered expansion wave. Ludwig Prandtl and his student Theodor Meyer first worked out a theory for centered expansion waves in 1907-1908, and hence such waves are commonly denoted as Prandtl-Mayer expansion waves.

17

1

μ2

μ2

θ

Forward Mach Line

μ1μ2

M1>1

P1

ρ1

T1

Rearward Mach Line

M2

P2

ρ2

T2

θ

μ1

μ1

2

Steps in solving expansion waves problems:

1. sin μ1 =

;

sin μ1 =

sin μ1 =

2. To find M2, use Table (Function for calculations of Expansive Flow).

[f (μ1)] = [f (μ1)]

3. [f (μ2)] = [f (μ1 )] θ

To find μ2:

μ2 = f [f (μ2)]

4.

5. cos μ1 =

cos μ2 =

6.

Where: [ g(μ2) ] = f(μ2)

[ g(μ1) ] = f(μ1)

18

From formulas of Bernoulli’s Theorem of compressible flow using M1 and M2.

7.

Where: [ h(μ2) ] = f(μ2)

[ h(μ1) ] = f(μ1)

Problems:

1. Using the exact theory method, find the final Mach number, pressure and density in the expansion case of an initial Mach number of 1.75 and a deflection of 8°. Assume SSLC.

2. Find all aerodynamic and thermodynamic conditions on both sides of expansion wave. Supersonic stream is at M = 1.6 expanding through an angle of 8°. Initial pressure and temperature are at 10 psia and 20°F, respectively.

19

20

FOR CALCULATIONS OF EXPANSIVE FLOW

μ f(μ) g(μ) h(μ)0 0.00° 0.00000 0.000001 5.00° 0.00000 0.000002 9.98° 0.00000 0.000003 14.92° 0.00000 0.000024 19.81° 0.00000 0.000095 24.63° 0.00001 0.000266 29.36° 0.00003 0.000617 34.00° 0.00009 0.001268 38.53° 0.00020 0.002329 42.94° 0.00043 0.0039210 47.22° 0.00081 0.0062111 51.37° 0.00143 0.0093112 55.37° 0.00237 0.0133213 59.23° 0.00370 0.0183214 62.95° 0.00552 0.0243815 66.52° 0.00791 0.0315316 69.94° 0.01095 0.0397717 73.21° 0.01469 0.0490618 76.34° 0.01919 0.0593719 79.34° 0.02446 0.0706220 82.19° 0.03053 0.0827321 84.91° 0.03739 0.0956222 87.50° 0.04502 0.1091823 89.96° 0.05338 0.1233024 92.30° 0.06243 0.1378925 94.53° 0.07211 0.1528526 96.64° 0.08236 0.1680827 98.65° 0.09313 0.1835028 100.56° 0.10434 0.1990129 102.36° 0.11592 0.2145530 104.07° 0.12780 0.2300531 105.70° 0.13994 0.2454432 107.23° 0.15225 0.2606833 108.69° 0.16468 0.2757134 110.07° 0.17718 0.2905135 111.37° 0.18970 0.3050336 112.61° 0.20220 0.3192437 113.77° 0.21462 0.3331438 114.88° 0.22694 0.3466939 115.92° 0.23913 0.3598840 116.90° 0.25115 0.3727141 117.83° 0.26297 0.3851742 118.71° 0.27459 0.3972543 119.54° 0.28598 0.4089444 120.32° 0.29712 0.4202645 121.06° 0.30800 0.4312046 121.75° 0.31862 0.4417747 122.40° 0.32896 0.4519648 123.02° 0.33901 0.4617949 123.59° 0.34879 0.4712550 124.14° 0.35827 0.4803751 124.64° 0.36746 0.4891452 125.12° 0.37635 0.4975753 125.57° 0.38496 0.5056754 125.99° 0.39327 0.5134555 126.38° 0.40130 0.5209156 126.74° 0.40904 0.5280757 127.08° 0.41650 0.5349358 127.40° 0.42367 0.54149μ f(μ) g(μ) h(μ)59 127.69° 0.43057 0.5477860 127.97° 0.43720 0.5537961 128.22° 0.44356 0.55953

62 128.45° 0.44965 0.5650163 128.67° 0.45549 0.5702464 128.87° 0.46107 0.5752265 129.05° 0.46639 0.5799666 129.22° 0.47147 0.5844667 129.37° 0.47631 0.5887468 129.51° 0.48091 0.5927969 129.64° 0.48527 0.5966370 129.75° 0.48941 0.6002571 129.85° 0.49331 0.6036772 129.95° 0.49700 0.6068973 130.03° 0.50046 0.6099174 130.10° 0.50371 0.6127375 130.16° 0.50674 0.6153676 130.22° 0.50956 0.6178177 130.27° 0.51218 0.6200878 130.31° 0.51459 0.6221679 130.34° 0.51680 0.6240780 130.37° 0.51881 0.6258081 130.39° 0.52062 0.6273682 130.41° 0.52224 0.6287583 130.42° 0.52366 0.6299784 130.44° 0.52489 0.6310385 130.44° 0.52593 0.6319286 130.45° 0.52678 0.6326587 130.45° 0.52744 0.6332188 130.45° 0.52791 0.6336289 130.45° 0.52819 0.6338690 130.45° 0.52828 0.63394

Approximation Method for Oblique Shock & Expansion Waves

This method provides the simple means of determining the change in flow conditions through oblique shock and expansion waves, particularly pressure distribution on airfoils to obtain the airfoil section characteristics. In this method, the deflection angle is taken as positive (+) for oblique shock wave (OSW) and negative (-) for expansion wave (EW).

A. Pressure Ration and Pressure Coefficient

Where:

= pressure ratio across OSW or EW

= pressure coefficient across OSW or EW

B. Pressure Coefficient Determination

1. First-Order Approximation – Sometimes referred to as Linear of Ackeret Theory in which θ is in first order.

= C1θ where:

θ = deflection angle (rad)

2. Higher-Order Approximation – In higher-order approximations, higher-order terms are added to the first-order term.

a. Second-Order Approximation

= C1θ + C2θ2 where:

b. Third-Order Approximation – For OSW, a coefficient D is introduced to correct for the non-isentropic flow losses across OSW.

= C1θ + C2θ2 + C3θ3 ------------- Expansion Wave

= C1θ + C2θ2 + (C3-D)θ3 ------------- Oblique Shock Wave

Where:

For Air, γ=1.4

For Air, γ=1.4

Problem:1. Using the Third-Order Approximation, find the pressure acting on the upper and lower

surface of a flat plate inclined at an angle of 10° toward the airflow at initial Mach number of 2 and freestream pressure of 1 atm.

c. Application of Pressure Coefficient – Pressure Coefficient can be used to determine important aerodynamic characteristics of airfoil sections. A practical approach is to use the second-order approximation which is more accurate than the first-order method and much simpler than the third-order method. As an example, the Normal

Force Coefficient of a Thin Plate will be considered here.

Where:

= C1θ + C2θ2 ------ OSW

= C1(-θ) + C2(-θ)2

= -C1θ + C2θ2 ------ EW

Where:CN = Normal Force Coefficientθ = Deflection Angle or Angle of Attack (radians)M1 = Mach number before Flat Plate

Problem;1. A two-dimensional thin flat plate is set against an airstream initially at 1,500 meters altitude

at 2,000 mph at an angle of 15°. Determine the normal force coefficient and pressure at the lower and upper surface of the plate.

Airfoil Characteristics

Two-dimensional Characteristics

Basic Aerodynamic Characteristics of Wings:

1. Lift – is that component of force which is normal to the direction of the free-stream an infinite distance ahead of the airfoil.

Where:

=

2. Drag – is that component of force parallel to the free-stream direction ahead of the wing.

Where:

=

Three Components of Total Drag

a. Form or pressure drag of “wave drag” – is the airwise force resulting from the pressure distribution when the wing is at the angle of attack at which no lift is generated.

b. Skin Friction – is the force created by the tendency of the layer of air next to the surface of the body to cling to the layer next to it until the free-stream velocity is reached. This shearing action creates a drag on the surface which is a function of the viscosity, velocity, density and the type of flow (laminar or turbulent) of the air, as well as the airwise surface dimension.

c. Drag due to lift or “drag due to normal force” – it is the component of the normal force which is parallel to the direction of the free-stream. It exists only when the airfoil is in an attitude to supply lift and hence is usually given the name shown at the heading. This force is called induced drag in the subsonic case but is not so called in the supersonic case because the type of flow over the lifting surface is of a different character. The term “induced drag” is sometimes used to describe the drag created in the region of the tips of rectangular wings producing lift in supersonic flow, however.

3. Axial force – is that component of force parallel to the chord or axis of the wing or body and is equal to the form drag plus skin friction at zero degree angle of attack.

4. Normal force – is the component normal to chord line of the wing.

5. Resultant force – is the force representing the resultant of the addition of all local aerodynamic and viscous forces on the wing from which lift and drag force are resolved.

6. Moment – is the resultant of the moments of all forces on a body about some chordwise reference point.

Forces Acting on Airfoil at Supersonic Speeds

L = NY – AY ; L = Ncosα 2 AsinαD = NX – AX ; D = Nsinα + Acosα

At α = 0:

L = N ; CL = CN

A = D ; CA = CD

Aerodynamic Characteristics of a Two-dimensional Symmetrical Double-Wedge Airfoil

= C1θ + C2θ2 ------ second-order degree of accuracy

Where:

= pressure change across an oblique compression or an expansion wave

θ = local angle of attack in radians between the surface and free-stream Mach number.M = free-stream Mach number.

Notes* θ = (+) when the free-stream is deflected toward the surface. θ = (-) when it is away from the surface.

The values of θ to be used in the pressure equation for each surface are as follows:

θ1 = -α + β

θ2 = -α – βθ3 = α + β

θ4 = α - β

Where:α = angle between free-stream direction and chordline of wing

away form the surface; α is bigger than β

opposite, open to the left

β = semi-vertex angle of leading and trailing edges.

Pressure Difference over the Front and Rear Halves of the Airfoil at Angle of Attack

Pressure Distribution over Symmetric Double-Wedge Airfoil

= -

= 2C1α + 4C2αβ

= -

= 2C1α - 4C2αβ

The normal force coefficient CN = will be the total difference between the upper and lower pressure difference (divided by 2 to retain the coefficient form based on the total plan-form area).

, α in radians

Where:

= pressure coefficient parameter

Total Chordwise Force Coefficient

= 2C1β

------- Based on the maximum thickness of the airfoil

Where:= from drag coefficient

= tanβ

For small angle β, tanβ β, in radians

------ Based on planform area

Drag due to normal force/due to lift

D = N sinα + A cosαCN sinα + CA cosα Note: CA =

For small angles of attack:

CA cosα is very small and sinα α in radiansSince,

Therefore,

Additional drag due to skin friction

Depends on the type of strength of shock waves created on the RN whether or not the flow is laminar or turbulent.

Maybe approximated by the use of coefficients developed by Blasius (laminar) and Von Karman (turbulent).

Total two-dimensional (sectional) drag of this airfoil

+ + = +

Where:β and α are in radians and is from the graph against R for turbulent and laminar flow.

Coefficient of moment about the mid-chord

- By summating the lift of each surface times the distance to its center of pressure divided by the chord length, to retain non-dimensionality.

Center of Pressure (C.P.) distance at or ahead of the mid-chord point.

- Moment divided by the normal force

Note: For a symmetrical double wedge airfoil, βrad. τ; βdeg. 57.3τ

Three General Contours that include the most practical types

The form drag coefficient for any given type of a cross-section may be expressed directly as a function of the thickness ratio:

Where:

Τ = the ration of maximum thickness to chord length, .

K1 = a constant which depends only on the cross-sectional shape.

Type K1

Double-Wedge 4Biconvex 5.33Modified double-wedge (a=⅓) 6Modified double-wedge – General

Note: a = is the fraction of chord length of the wedge shape at each end.

Optimum Angle of Attack and Maximum ratio

The normal force-drag ration is determined, rather than the lift-drag ratio, because the simplicity of the normal force expression as compared with that of the loft. Form nominal values of lift-drag ratios, i.e., 5 occurring at optimum angles less than α=10 degrees, the lift-drag ratio will be no more than 5 percent lower than the normal force-drag ratio. The optimum angle of attack for

highest ratio will also be slightly higher than that for Dmaximum .

The optimum angle of attack for maximum is found by differentiating the expression

to determine the slope as a function of angle of attack and setting this expression equal to zero

because the optimum angle occurs where is a minimum.

Since,

The expression for optimum angle of attack is related to the maximum lift-drag ratio by the relation:

where: α is in radians

Moment and Center of Pressure

Where:

Shape

Double-Wedge

Biconvex

Modified double-wedge (a=⅓)

Modified double-wedge – General (1-a)τ

Problems:

1. Find the section characteristics of a symmetrical double-wedge airfoil of 6% thickness ratio at S deg. Angle of attack is a supersonic stream of Mach no. M=1.8. Total skin friction = 0.0053. All coefficients are converted to degree measure in calculations.

2. Find the sectional drag coefficient, optimal angle of attack, and maximum normal force-drag ratio for a 6% biconvex airfoil at M=1.8, = 0.0053.

For the double-wedge