Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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21.7 Laser Doppler Anemometry 715

∆f

= lim

N→∞



k=1

)

− f

∆t



k=1

∆t

= lim

N→∞



k=1

)

− f

= <f

(21.116)

In accordance with this, the time-averaged and the ensemble-averaged prop-

erties of a ﬂow agree with one another in the special case that constant

averaging time intervals are chosen. It is therefore necessary to process the ob-

tained Doppler signals accordingly, so that one corresponding Doppler signal

is attributed to a particular constant time interval ∆t.

The above explanations of the behavior of LDA signals have shown that

the laser Doppler signals occur at irregular time intervals, so that irregular

scanning intervals are given by the Doppler signals. When this is not taken

into account, the time-averaged Doppler frequency and the corresponding

ensemble-averaged value can diﬀer from one another. This is not an error

of the measurement technique, but a characteristic of the chosen averaging

process. The diﬀerences that occur are clear and follow from the deﬁnitions of

the mean values of ensemble and time. Hence, there are no principle problems

to measure biasing free averaged velocities in ﬂuid ﬂows.

21.7.4 Execution of LDA-Measurements:

One-Dimensional LDA Systems

In the preceding sections on LDA signal processing, LDA measurements were

presented without entering into the data obtained by the evaluation, in order

to obtain from the measured Doppler frequencies the desired information

on the ﬂow ﬁeld. It was shown that laser Doppler anemometers are linear

velocity measurement value sensors with a frequency response that is given

by the following equation:

, (21.117)

where λ corresponds to the wavelength of the chosen laser radiation,

the momentary velocity vector and (n)

to the transformation vector of the

anemometer. For a stable coordinate system x

,thetwovectorsoftheabove

equation can be expressed in the following form:





=(U

)and(n)

=2sinϕ (cos α

, cos α

), (21.118)

where ϕ is half the angle between the light beams and α

, α

and α

are the

angles which the vector (n)

forms with the coordinate axes (Fig. 21.53). The

momentary velocity components and the momentary signal frequency can be

expressed as follows:

= U

+ u

= f

+ ∆f

(21.119)

716 21 Introduction to Fluid-Flow Measurement

Fig. 21.53 Important parameters for the evaluation of LDA signals in measurements

with one-dimensional LDA measurement systems

By combining the above equations, the following relation results:

+ ∆f

2sinϕ

cos α

+ U

cos α

+ U

cos α

+ u

cos α

+ u

cos α

+ u

cos α

)

(21.120)

This equation represents the basic relation for the evaluation of ﬂuid

mechanical quantities from frequency measurements of laser Doppler signals.

On carrying out a time average for (21.120), one obtains the following

basic equation for the determination of the three velocity components U

, U

and U

. To be able to measure all three components with a one-dimensional

LDA system, measurements in three diﬀerent directions, α

, α

,andα

are

required:

2sinϕ

cos a

+ U

cos a

+ U

cos a

). (21.121)

On deducing (21.120) from (21.121), one obtains the equation for the diver-

gence of the frequency from the averaged frequency. With this an equation

can be derived for the standard divergence from the Doppler frequency ∆f

∆f

4sin



cos

+2( ¯u

cos a

+¯u

cos a

+¯u

cos a

)]

(21.122)

The measurement of the standard divergences of the signal frequencies for

six diﬀerent directions of the sensitivity vector (n

)

(j =1, 2,...,6) makes

possible the evaluation of all correlations

of second order.

Similar evaluating equations can be derived for the correlations of higher

order:

∆f



2sinϕ



cos a

+ u

cos a

+ u

cos a

]

The complexity of the evaluating equations reduces, when preferential direc-

tions are given for the transformation vectors (n)

, e.g. parallel to the x

-axis

(n)

=(1, 0, 0):

λf

2sinϕ

;

∆f

4sin

;

∆f

sin

(21.123)

References 717

With this, the evaluation of the measurements which are carried out with a

one-dimensional optic, is similar to the measurements with a single hot wire.

However, laser Doppler anemometers are characterized by a precise Cosine-

law response behavior and therefore the evaluating equations are easier. More

details about Laser Doppler anemometry can be found in a recent published

book by Albrecht et al. [21.10] with details about the physics of the measuring

technique and good descriptions about LDA-applications.

References

21.1. Hinze JO (1975) “Turbulence”, 2nd edition, McGraw-Hill, New York

21.2. Hanson S (1976) “Visualization of alignment errors and heterodyning con-

straints in laser Doppler velocimeters”. Proceedings of the LDA Symposium,

Copenhagen, 1975, pp. 176–182

21.3. Buchhave P, Delhaye JM, Durst F, George WK, Refslund K, Whitelaw JH

(eds.) (1976) “The accuracy of ﬂow measurements by laser Doppler methods”.

Proceedings of the LDA Symposium, Copenhagen, 1975.

21.4. Durst F, Stevenson WH (1979) “Inﬂuence of Gaussian beam properties on

laser-Doppler signals”, Applied Optics, 18, pp. 516–524

21.5. Goldstein RJ, (ed.) (1983) “Fluid Mechanics Measurements”, Hemisphere,

Washington, DC

21.6. Wiedemann J (1984) “Laser-Doppler-Anemometrie”, Springer, Berlin

Heidelberg New York

21.7. Ruck B (1987) “Laser-Doppler-Anemometrie”, AT Fachverlag, Stuttgart

21.8. Durst F, Melling A, Whitelaw JH (1987) “Theorie und Praxis der Laser-

Doppler-Anemometrie”, G. Braun, Karlsruhe

21.9. Eckelmann H (1997) “Einfuehrung in die Stroemungsmesstechnik”, BG

Teubner, Stuttgart

21.10. Albrecht HE, Borys M, Damaschke N, Tropea C (2003) “Laser Doppler and

Phase Doppler Measurement Techniques”, Springer, Berlin Heidelberg New

York

Index

Aerostatics, 183–187, 189

buoyancy, 188–190

stability, 191, 192

Axi-symmetric ﬁlm ﬂow, 376–378

Balance considerations, 73–75

Bernoulli equation, 130, 133, 134, 144,

145, 231, 232, 234, 241, 243,

254–258, 302, 304, 307, 321, 326,

341–344, 507

Blasius solution, 470–474, 479, 481

Boundary layer, 481

equations, 12, 463–493, 588, 641–651

ﬂows, 209, 210, 463–493, 520, 521,

578, 641, 647

with heating, 641–651

Bubble formation on nozzles, 175–183

Channel ﬂow, 207, 369–371, 388–390,

407, 417–426, 489–492, 514–516,

519, 550, 568–572, 575, 578–585

Characteristic function, 516, 527, 537,

538

Classiﬁcations of instabilities, 44, 282,

517–519

Complex functions, 42–47

Complex numbers, 11, 36–43, 45

Computation

laminar ﬂows, 614, 615

turbulent ﬂows, 616–626

Connected containers, 163–168

Conservation laws, 73, 75

for chemical species, 74, 113, 114

for energy, 74, 113, 114

for mass, 74, 113, 114

for momentum, 74, 113, 114

Continuity equation, 74, 115–119, 135,

136, 148, 150, 153, 197, 200, 209,

221–224, 231, 232, 234, 251–253,

257, 258, 260, 261, 276–279,

313–316, 332, 353, 356, 362–366,

432, 466, 469, 471, 509, 513,

543–546, 558, 560, 561, 568, 592,

603, 607–609, 616, 630, 631, 641,

642, 648

Continuum mechanics, 51–55, 57–59, 61,

62, 64, 116, 678

Convection inﬂuence, 459–461, 463–465

Converging channel ﬂow, 489–492

Coordinate systems, 17–20, 32–36,

120–124, 135–142, 184, 361–364,

443, 446, 471, 598, 635, 685, 691

Corner and sector ﬂows, 284–288

Correlations of turbulence ﬂuctuations,

538–542

Creeping ﬂuid ﬂows, 431–433

Crocco equation, 142, 146, 147

Deformation operation, 104–107,

110–112

Diﬀerent wave motions, 311, 312

Diﬀerential equations, 75, 87, 89, 90,

92, 93, 96, 113–151, 275–286, 313,

316–318, 321–324, 332, 351, 365,

367, 370, 373–377, 379, 383, 384,

386, 388, 394–397, 400, 402–404,

406–409, 414–419, 422–425, 430–

432, 447–449, 451, 454–456, 464,

465, 468–472, 480–484, 487, 488,

490, 491, 496, 501–515, 557–559,

576, 577, 579, 587–590, 595, 596,

632, 634, 638–642, 644–646, 648,

653, 690

719

720 Index

boundary layers, 210, 464, 465, 468,

470, 490, 491, 520, 588, 642, 645,

646

chemical species, 150

dimensions forms, 638

ﬂuid ﬂows, 8, 123, 199, 208–211, 332,

386, 431–433, 642–648, 653

heat transfer, 197–204, 627, 628, 638,

642

molecular transport, 197, 198

numerical solutions, 587–590, 595,

596, 600, 612, 614

potential ﬂows, 275, 281, 293

wave motions, 313, 316–318, 321, 323,

324

Diﬀerential operators, 32–36

Diﬀusion

inﬂuence, 459–461

of vortex layer, 399–402

Dimensional analysis, 212–218

Dipole generated ﬂow, 291–294, 296

Discretization

in space, 600–611

of derivatives in space, 589

of derivatives in time, 589, 611–613

Divergence operator, 28

Dynamic pressures, 259, 343, 655, 660,

662

Eddy viscosity models, 560, 562, 564,

565, 573, 576

Electronic systems, 705

Elementary complex functions, 42, 43

Euler’s turbine equations, 242–244

Exit velocity of nozzles, 231, 232

Fanno relation, 351

Field variables, 11, 23, 24, 27, 54, 75, 78,

83, 84, 115, 118, 146, 221

Film ﬂow on an inclined plate, 372–376

Film ﬂows with two layers, 386–388

Finite diﬀerence discretization, 589,

595–598

Finite volume discretization, 598

Flat plate boundary layer, 209, 470, 473,

474, 479, 514, 521, 645

Flow

forces on bodies, 302–307

with heat transfer, 344–350, 627–633,

635, 644

between vertical plates, 633

Flow around a cylinder

with circulation, 296, 306, 307

without circulation, 297

Flow between two cylinders, 383–385

Fluid ﬂow between plates, 369–371

Fluid ﬂow measurements, 210, 653, 669,

704

Fluid Mechanics

analytical, 4, 7, 8, 11, 73

development of, 1–14

experimental, 5, 7, 8, 655, 660, 664

historical developments of, 9–14

numerical, 5, 8, 9, 13, 14, 83, 591

signiﬁcance, 1–4

sub-domains, 4–8, 12, 85

Force

on bodies, 302–307

on periodical grid of plates, 240–242

on turbine plate, 239, 240

Formation of shock waves, 338

Free ﬂuid surface, 160, 168, 169

Free shear layer, 480, 481

Fully developed ﬂow in channels, 630

Gas dynamics, 8, 70, 331–360

Gauss complex number plane, 38, 40

Gauss law of integration, 31, 32

General transport equation, 55, 591–593,

595, 597, 614, 618, 620

Geometric similarity, 193

continuity equation, 197–200, 209

mechanical energy equation, 198

momentum equation, 197–199, 205,

209

thermal energy equation, 194, 198

Gradient operator, 27, 28

Gravitationally caused instabilities, 505

Hagen-Poiseuille ﬂow, 379–383

Heat transfer, 196–198, 201–204, 211,

344, 347, 627–630, 637, 641, 642,

645, 647, 649, 654, 664, 667, 670,

671, 676–681, 684

Hot wire anemometry, 664, 665, 678,

679, 687

cooling laws, 676

probes, 672

static calibration, 680

supports, 672

Hydrostatics, 10, 153–156, 158, 161, 183,

188, 190

Ideal gases, 23, 50, 58, 59, 61, 62, 64–66,

76, 79–81, 116, 123–125, 127, 129,

133, 134, 183, 184, 198, 262, 265,

310, 313, 314, 331, 332, 334, 345,

347, 349, 354, 355, 359–361

Index 721

Important potential ﬂows, 299

Incompressible ﬂows, 257–273

complex function, 280–282

potential ﬂows, 277, 280

potential function, 276–282

stream function, 98, 275–280

Integral laws, 29–32

Integral properties, 474–480

Jet deﬂection at an edge, 236

Jet ﬂow, 481–486

Kinematic similarity, 194, 198

Kinetic energy of turbulence, 550, 555

Laminar free jet, 481–486

Laminar-turbulent transition, 12, 495,

519–522

Laplace operator, 26–28, 34

Large Reynolds number ﬂows, 463–492

Laser Doppler anemometry, 7, 13, 558,

559, 565, 653, 655, 694–696, 701,

709, 710, 713, 717

Length scales, 207, 208, 210, 553–556,

562–564, 576, 579, 624

Longitudinal waves, 310, 313–318

Lubrication ﬁlms, 433, 434, 436

Mach cone, 335, 337, 338

Mach lines, 335

Mass conservation, 73–75, 99, 100,

114–117, 221–224, 243, 256–258,

344, 345, 607, 616

Mean ﬂow ﬁeld, 125, 383, 433, 543–546,

548–554, 558, 564, 572–574

Mechanical energy equation, 128, 130,

144, 145, 221, 225, 226, 228, 245,

255, 256, 309, 546, 547, 560, 627

Mixing process in pipe, 237

Molecular heat transport, 83

Molecular mass transport, 83

Molecular momentum transport, 69, 83,

123, 125, 140, 141, 148, 200, 209,

225, 253, 354, 371, 385, 388, 394,

397, 398, 401, 453, 459, 480, 486,

588

Molecular properties, 51, 52, 54, 64, 67

Momentum

equations, 85, 119, 120, 123, 127, 128,

130, 137–139, 144, 197–199, 209,

253, 275, 277, 303, 313, 338, 356,

362, 363, 365, 366, 459, 496, 509,

546, 617, 618, 628, 634, 637, 646,

649, 653

on inclined plane plate, 234–236

on plane vertical plate, 232–234

Motion of ﬂuid elements, 85–89

deformation, 104–107, 110–112

divergence, 101–103

path lines, 86–89

relative motions, 108–112

rotation, 104–107, 110, 111

streak lines, 90–94

stream function, 98–101

stream lines, 94–101

translation, 104–107, 110–112

Natural convection, 191, 633–636, 638,

640, 647–649

Navier-Stokes equations, 12, 123–128,

136–144, 146, 183, 277, 312, 361,

381, 430, 446, 449, 450, 464–466,

468, 476, 497, 501, 505, 509, 513,

542, 544, 545, 557, 560–562, 565,

592, 614, 641, 653

Newton’s second law, 119–123

Newtonian ﬂuids, 4, 8, 55, 69, 124, 127,

137, 197, 201, 206, 221, 364, 369,

381, 544, 546, 562, 593, 650

Non-linear propagation, 338–341

Numerical solutions, 5, 6, 8, 13, 473,

476, 501, 516, 526, 559, 560, 562,

587, 588, 590–594

One-dimensional ﬂows and viscous

ﬂuids, 393, 394, 396, 397, 406, 426

One-equation models, 563, 573, 576

Optical systems, 696, 697, 700–706, 708

Orthogonal coordinates, 32–36

Oscillating ﬂuid ﬂows, 414–416

Outﬂow from containers, 230, 231

Partial derivative, 85, 280, 394, 400

Pipe ﬂow, 200, 252, 258, 344–352, 355,

379–383, 385, 407, 424, 426, 572,

578, 584

Plane Couette ﬂow, 366–369

Plane plate boundary layer, 641–644

Plane progressive waves, 325–329

Plane standing waves, 323–325

Potential ﬂows, 275–308

Potential vortex ﬂow, 288–291

Power of ﬂow machines, 245–247

Pressure gradient driven ﬂuid ﬂows,

417–426

Pressure measuring instruments,

163–168, 657, 661, 662

722 Index

Probability density, 527, 530–538

Properties of hot wires, 654, 667–672

Rankine–Hugoniot equation, 355–360

Rayleigh relation, 351–355

Reynolds equations, 542, 544–546, 557,

558, 562, 568, 569, 575, 579, 653

Roller bearing, 438–443

Rotating containers, 187, 188, 432

Rotational motion of cylinder, 451–453

Rotation operator, 27–29

Rotation of sphere, 443–445

Rules of time averaging, 120, 528–530

Scalars, 16, 17, 24

Similarity solution, 644–647

Similarity theory, 193, 197, 198, 212, 678

Small Reynolds number ﬂows, 429–431,

447, 450, 454, 458, 461, 463, 496,

556, 557, 560

Solids and Fluids, 49–51

Sound waves in gases, 313–318

Source or sink ﬂows, 288–291

Spectra of turbulent ﬂows, 538–542

Stability of Flows, 517, 519, 520

Stagnation pressure, 662–664

Starting ﬂow in channel, 417, 418,

420–422

Starting pipe ﬂow, 422–426

Static pressures, 655–660

Statistical considerations, 528

Stereographic projection, 41, 42

Stokes’ ﬁrst problem, 397, 399

Stokes’ law of integration, 31

Stokes’ second problem, 414

Stream line, 94–98

Stream tube theory

Bernoulli equation, 254–258

continuity equation, 251–253, 257,

258, 260, 272

momentum equation, 253, 254, 258,

259

total energy equation, 256

Substantial derivative, 26, 27, 76, 84, 85,

117

Substantial variables, 27

Surface tension, 168–172, 196, 310, 326,

329, 505

Surface waves, 318, 320–322, 327, 330,

517

Taylor instability Orr-Sommerfeld

equation, 512

Tensors

ﬁrst order, 15, 17–21, 24

higher order, 23, 630

second order, 15, 21–23

zero order, 15–17, 24

π-Theorem, 212–214

Theory of Laser Doppler anemometry,

694

Thermal energy equation, 128, 130, 132,

133, 147, 228, 229, 257, 313, 561,

627

Thermodynamic considerations, 76–81,

146

Time dependent ﬂows, 393, 394, 426

Time scales, 333, 538–542, 553, 554, 571,

572, 650, 651, 710

Time scales of turbulence, 538, 539

Total derivative, 85

Transitional boundary layer ﬂows, 519

Translatory motion

of cylinder, 453–459

of sphere, 445–451

Transport equation for vorticity, 143,

459, 593, 614

Transport processes, 1, 12, 15, 55–58,

62, 65, 194–196, 202, 393, 464, 469,

495, 496, 523, 565, 566, 612

Transversal waves, 310, 318

Treatment of source terms, 613

Turbulence

ﬂows, 12, 13, 85, 385, 496, 497, 499,

500, 523–585, 616–620, 624, 653,

654, 661, 662, 673, 685, 687, 691,

713

measurements, 685–694

models, 14, 555, 557, 558, 560,

562–565, 573–578, 617–620, 653

wall boundary layers, 572, 578, 582,

584

Two-equation models, 563, 564

Two-phase plane channel ﬂow, 388–390

Unstable Flows, 495, 497, 501, 517

Vectors, 15, 17–26, 28, 30–35, 39, 57, 86,

94, 95, 101–105, 109, 110, 143, 144,

147–149, 154, 158, 168, 222, 223,

276, 277, 336, 355, 526, 686, 687,

695, 696, 715, 716

Index 723

Velocity scales, 430, 554, 555, 567

Viscosity of Fluids, 69–73

Wake ﬂow, 486–489, 500

Wall boundary

conditions, 579, 615, 616, 620

layers, 210, 464, 474–479, 514, 520,

559, 574, 578–585, 654

Wave motions, 309–330

Wave propagation, 312, 318, 337, 338

Zero-equation models, 576