Discrete and Continuous Spectrum Trailing Line Vortex

Blunt Trailing Edge

For the blunt trailing edge in the subsonic speed range, the base flow is dominated by the periodic formation and shedding of eddies into the wake.

From: Separation of Flow , 1970

Wake Flow

PAUL K. CHANG , in Separation of Flow, 1970

III Wake behind a blunt trailing-edge at subsonic and transonic speeds

Recent investigations of wakes behind a blunt trailing-edge are shown in Fig. 13. Nash et al. [13] indicate that the static pressure along the center line of the wake falls rapidly away from the base, reaching a minimum at a distance downstream equal to about half the trailing-edge thickness, as seen in Fig. 14.

FIG. 13. Diagram of basic blunt trailing-edge model [13]

FIG. 14. Static pressure on center line of wake [13]

For comparison, the static pressure distribution behind a flat plate [14] is shown in Fig. 14.

The low pressure "trough" shown in Fig. 14 has also been noticed by other investigators [14, 15, 16]. This trough appears to coincide with the point at which the vortices form. The value of pressure coefficient C_p in the trough seems to decrease with an increasing degree of bluffness of the section; hence, presumably with increasing strength of the vortex street. For example, from Fig. 14, the minimum value of C_p is about –1·0, compared with –1·4 for a circular cylinder [16] and with –2·3 for flat plate normal to the stream [14]. Downstream of the trough, the static pressure rises again and levels out at some value below the ambient static pressure. The presence of this pressure plateau differing from the progressive increase of static pressure downstream of the wake center line indicates the existence of a vortex street. Theoretically, it may be shown that the static pressure along the center line of a vortex street is lower than free-stream static pressure by an amount depending on the strength of the street. With the increase of Mach number, reaching the flow regime in which the trailing shock develops, the point at which the vortices form is pushed downstream, replacing the low-pressure trough. At Mach numbers above transition to "steady" base flow (for model sketched in Fig. 13, M = 0·975), the constant pressure region downstream is observable, which indicates the establishment of semi-dead air, and the static pressure in the wake approaches the free-stream value without any intermediate plateau. Thus although some evidence of the persistence of discrete eddies in the wake at supersonic speeds exists, the reduction of their strength is such that the wake displays few of the characteristics of fully developed vortex street. For the study of wake growth, first the wake thickness h′ at a given station is defined as the distance between points on opposite sides of the center line at which the value of the parameter

$\frac{H_0-H}{H_0-P^{\infty }}$

becomes equal to half the value on the center line, at that station where H is the Pitot pressure, H ₀ is the stagnation pressure, and p _∞ is the ambient static pressure, as shown in Fig. 15. The variation of wake thickness is almost the same as the base pressure coefficient through the regime in which periodic effects predominate, as seen in Fig. 16. The transition from the predominately unsteady flow regime to that of the steady one at M = 0·975 is marked by a decrease of wake thickness.

FIG. 15. Definition of h′ [13]

FIG. 16. Relation between base pressure, wake thickness, and vortex-shedding frequency [13]

In Fig. 16 are plotted the Strouhal numbers S and S′ defined by

$S=\frac{nh}{u_{\infty }}\text{and}S\text{'}=\frac{nh\text{'}}{u_{\infty }}=\frac{h\text{'}}{h}S$

where n is shedding frequency and h the base height. In the transition region from unsteady to steady flow, the shedding frequency increases discontinuously. The free stream line of various body shapes may be determined by using the hodograph method as shown by Birkhoff and Zarantonello [17].

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Evaluation of Parallelized Unstructured-Grid CFD for Aircraft Applications

Takeshi Fujita , ... Takashi Nakamura , in Parallel Computational Fluid Dynamics 2002, 2003

4.3 Navier-Stokes computation for regional jet airplane

The NS equations are solved for a regional jet airplane as an application of large scale computations. This model has a blunt trailing edge wing. This unstructured hybrid mesh contains 2,180,582 nodes, 3,839,284 tetrahedrons, 2,943,184 prisms and 10,577,008 edges. The NS computation requires about 4GB memory. Figure 10 shows the pressure distribution on the surface. The comparison of the scalability result with NEC TX-7/AzusA and NEC SX-4 is shown in Fig. 11. Each two results show extremely good scalability. And NEC SX-4 32 PE's achieves 10 GFLOPS. The computation, which takes over 3 days without parallelization, is finished in less than 3 hours by parallel execution using 32 PE's.

Figure 10. Pressure Distribution of a Regional Jet Airplane with Blunt Trailing Edge

Figure 11. Comparison of Speedup Result for Regional Jet Airplane

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URANS Computations of Shock Induced Oscillations Over 2D Rigid Airfoil: Influence of Test Section Geometry

M. Thiery , E. Coustols , in Engineering Turbulence Modelling and Experiments 6, 2005

4.1 Mesh convergence

Mesh convergence was performed for steady flow conditions, α  = 2.5°. A particular attention was paid to the refinement of the blunt trailing edge (relative thickness 0.5% of chord length), of the longitudinal discretization of the airfoil suction side, and of the transverse discretization of the boundary layers and wake, as all these elements might interfere during SIO cycles.

The first grid 'CH₀ fine' was generated with a rather large number of discretizing points to be able to remove one point over two in each direction and generate the 'CH₀ coarse' grid. To complete the grid refinement, the 'CH₀ intermediate' grid was created by refining the 'CH₀ coarse' grid in both directions. The characteristics of these grids can be found in Table 1. The Richardson interpolate value of lift coefficient (Slater, 2004) was determined from these three grids. The 'CH₀ coarse' grid predicted a lift coefficient 1.1% smaller than the Richardson one, allowing to assume the grid convergence. Then, an iterative process on finer constraints than the lift coefficient (e.g. Reynolds stress slope in the wake) was performed and led to the generation of the CH₄ grid with 309 points along the airfoil and 65 points in the blunt trailing edge, which was chosen for unsteady computations.

Table 1. Grid characteristics and lift coefficient obtained at α  =   2.5° with the "2D inf." approach.

	Airfoil	Δy+	Wake	Δx +	N (total pts)	CL
CH0 coarse	317   ×   129	~   0.4	121   ×   305	~   2	77798	0.95733	(1.1%)
CH0 interm.	349   ×   149	~   0.4	129   ×   371	~   2	99860	0.9603
CH0 fine	633   ×   257	~   0.2	241   ×   609	~   1	309450	0.96614
Interpolate	-	-	-	-	∞	0.9683
CH4	309   ×   133	~   0.4	105   ×   329	~   2	75642	0.9587	(1%)

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Blade Tip Flow and Noise Prediction by Large-Eddy Simulation in Horizontal Axis Wind Turbines

O. Fleig , ... C. Arakawa , in Engineering Turbulence Modelling and Experiments 6, 2005

Validation of flow solution – NACA0012 blade section flow at high Reynolds number

The NACA0012 blade section in an incident flow at an angle of attack of 5 and 6 degrees with a Reynolds number of 2.87   ×   10⁶ and a Mach number of 0.205 is simulated with periodic spanwise boundary conditions. The blunt trailing edge geometry and the flow parameters are based on the experiment by Brooks et al. (1981). The chord length is 0.6096   m. Terracol et al. (2002) already simulated this flow using LES.

The computational grid is an O-grid consisting of 1713 points along the airfoil surface, 65 points perpendicular to the surface and 400 grid points in the spanwise direction, with a total of 44 million grid points. The suction side consists of 1630 grid points and the pressure side has 84 points. The grid spacing normal to the blade surface, Δy⁺, is approximately 1.0 along the entire blade surface. In the transition region, the streamwise grid spacing, Δx⁺, is approximately 50.0. The spanwise grid spacing, Δz⁺, takes a value of 50.0, resulting in a span which corresponds to 20   % of the chord length. The computational domain has a diameter of 16 chord lengths. The non-dimensional time step based on the free stream velocity is 2.0   ×   10^{-   4} c/U_∞.

Grid spacing effects and the effect of the Smagorinsky constant C_s can be studied to assess the effects of the numerical dissipation caused upwind scheme indirectly. This is done with respect to trailing edge wall pressure spectra.

Grid spacing effects

A streamwise grid spacing greater than Δx ⁺  =   50.0 in the transition region downstream of the leading edge on the suction side did not capture the instabilities associated with the transition from laminar to turbulent flow. The flow remained laminar and separated at mid-chord. The trailing edge surface pressure spectra did not agree well with experimental measurements. Increasing the number of grid points in the streamwise direction along the suction side beyond 1630 did not significantly alter the numerical results. A spanwise grid spacing in the order of Δz+   =   100.0 was found to be too coarse to properly capture transition to turbulent flow. Early laminar separation took place. A spanwise grid spacing smaller than Δz+   = 50.0 leads to the occurrence of transition effects near the leading edge. The location of laminar to turbulent transition as well as the vortex shedding frequency at the blunt trailing edge became consistent as the spanwise grid spacing was further decreased towards Δz ⁺  =   10.0.

Smagorinsky constant effects

The Smagorinsky constant C_s was assigned values of 0.00 (no eddy viscosity), 0.10, 0.15 and 0.20, and simulation results of the trailing edge wall pressure spectra were compared with experimental surface pressure spectra by Brooks et al. (1981) . The blade section has a blunt trailing edge with vortex shedding occurring at the trailing edge. The simulation crashed when applying no eddy viscosity model. This suggests that the eddy viscosity does indeed contribute to the dissipation and that not the entire dissipation is caused by the upwind scheme. Surface pressure spectra on the suction side at the trailing edge obtained by LES with C _s  =   0.10, 0.15 and 0.20 as well as experimental values are shown in Figure 1. The simulation results were obtained after calibrating the streamwise and spanwise grid spacing. They correspond to the following grid configuration: The streamwise grid spacing is set to Δx⁺  =   50.0 in the transition region. The spanwise grid spacing is Δz⁺  =   50.0. 400 grid points were taken in the spanwise direction, resulting in a span which corresponds to 20 % of the chord length. The LES simulation with C_s  =   0.10 yields a vortex shedding frequency of 2,500   Hz, which is lower than the experimental value of 3,300   Hz. The simulation with C_s  =   0.20 yields a shedding frequency of 4,400   Hz, which is much higher than the experimental value. The difference in simulation results due to varying values of the Smagorinsky constant suggests that the eddy viscosity does affect the dissipation, while the effect of the numerical dissipation caused by the upwind scheme is small. In this case the simulation results obtained with C_s  =   0.15 give results that are closest to the experimental measurements. The spectra illustrate a narrowband component due to the vortex shedding emerging out of a wideband continuum which is generated by turbulent boundary layers due to the bluntness of the trailing edge. The predicted vortex shedding frequency is slightly less than the experimental shedding frequency of 3,300   Hz. It can be said that there is favorable agreement between results obtained by the present flow solver and experimental results.

Figure 1. Surface pressure spectra on suction side at trailing edge, Experiment: Brooks et al. (1981)

It can be thought that the computational grid used in the present simulation is fine enough to resolve a remarkable part of the spectrum. By using a very fine grid and capturing the smallest eddies the effect of the numerical dissipation to annihilate the small scales and overwhelm the SGS contribution is reduced as well as is the dependency on the SGS contribution. Although the numerical dissipation term will not vanish even for very fine grids, using a grid as fine as the present one can partly compensate for the undesired effects of the upwind scheme. The present LES approach can be considered suitable for efficient computation of compressible LES for applications for a full wind turbine blade.

Surface pressure distribution

Figure 2 compares the experimental (Gregory et al. (1970)) and computational mean surface pressure coefficient for an angle of attack of 6 degrees obtained with C_s  =   0.15. Good agreement between computational and experimental results is observed.

Figure 2. Mean Surface pressure coefficient (α   =   6°), Experiment: Gregory and al. (1970)

Figure 3 shows the instantaneous pressure contours for an angle of attack of 5 degrees. Instabilities associated with the laminar-turbulent transition to turbulent flow can be seen on the suction side downstream of the leading edge. At approximately x/c  =   0.25, the instabilities disappear. The present configuration was found to have a grid spacing that is sufficiently fine to capture the unstable phenomena associated with the transition without applying any tripping mechanisms, without manipulating the eddy viscosity and without purposely refining the computational grid in the transition region.

Figure 3. Instantaneous pressure contours for α   =   5°

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Parallel Implementation of a Compact Higher-Order Maxwell Solver

Ramesh K. Agarwal , in Parallel Computational Fluid Dynamics 1998, 1999

6 RESULTS

As mentioned before, CEM code ANTHEM has been extensively validated by computing the TE and TM scattering from cylinders, airfoils and ogives by comparing the results with analytical solutions and MoM calculations. For the sake of brevity, here we present the results of parallelization of one case only, but similar performance has been achieved in other calculations. We consider TE scattering from a dielectric NACA0012 airfoil. The computational domain is shown in Figure 1 . The hatching on the airfoil identifies it as a PEC NACA0012 profile. Each of the two zones wraps around the PEC airfoil forming a periodic continuous boundary at the x axis. The chordlength of the PEC airfoil is 4λ₀ . The airfoil's thickness to length ratio is 0.12. A lossless dielectric coating of thickness t_coat   =   0. 1λ₀ surrounds the airfoil. The layer terminates as a blunt trailing edge having a thickness of 2 t_coat and extending t_coat beyond the trailing edge of the PEC airfoil. The radiation boundary has a diameter of 6λ ₀. The inner zone represents the lossless dielectric layer between the PEC and dielectric interface. The lossless media is characterized by ϵ_c  =   7.4 and μ _c  =   1.4. The outer zone represents freespace.

Figure 1. Computational domain: (a) Zones, (b) Grid, every second line shown, zone 1:13 x 13, zone 2: 513 x 35.

The two zone grid is shown in figure 1(b). Due to the density of the grid, every other grid line is plotted. For the chosen dielectric constants the speed of light is less than a third that of freespace. Due to the reduced local incident wavelength and interference patterns, increased grid density is required in the dielectric zone.

Computations are performed for TE scattering from the dielectric airfoil due to an incident wave at 45° angle to the chord of the airfoil. Figure 2 shows the comparison of bistatic RCS computed with the CEM code ANTHEM and the MoM code. Figure 3 shows the scattered field D_z phase contours obtained with the CEM code. These calculations have been performed on a single HP750 workstation with the computational parameters as shown in Table I.

Figure 2. Bistatic RCS: solid line, MoM solution, dashed line, ANTHEM solution.

Figure 3. Scattered D_z field phase

Table I. Performance on 1 workstation - HP750

Zone	Zone Dimensions	Number of Iterations	Memory (kBytes)
1	513x13	9540	3284
2	513x35

Cpu/iteration/grid point   =   7x10^{-   5} secs

Three-order of magnitude reduction in residuals

Tables I-III show the result of parallelization on one-, two-, and three-workstations. Several domain decompostition strategies are implemented and evaluated on multiple workstations. In Table II, three strategies are implemented on two workstations. In strategy 1, two zones are divided at the dielectric/freespace interface; as a result there is load imbalance. In strategy 2, two zones are equally divided but the split is in the η -direction. In strategy 3, two zones are equally divided with a split in ξ - direction. Strategy 2 yields the best performance. On a cluster of three workstations, as shown in Table III, a similar conclusion is obtained.

Table II. Performance on 2 workstations - 2 HP750

Domain Decomposition Strategy	Zone Dimensions		Number of Iterations	Ideal Parallel Efficiency	Actual Parallel Efficiency
	Zone 1	Zone 2
1	513x14	513x36	9880	95.4%	91.9%
2	513x25	513x25	9760	97.7%	93.2%
3	258x48	258x48	10360	92.1%	87.6%

Table III. Performance on 3 workstations-HP750, SGI Indigo, IBM RS/6000

Domain Decomposition Strategy	Zone Dimensions			Number of Iterations	Ideal Parallel Efficiency	Actual Parallel Efficiency (based on clock time)
	Zone 1	Zone 2	Zone 3
1	513x17	513x17	513x17	9910	96.3%	87.2%
2	172x48	172x48	172x48	10540	90.5%	82.0%

Note: ξ -grid line at η   =   1 represents the PEC NACA airfoil

ξ-grid line at η   =   13 represents the dielectric interface with freespace

ξ-grid line at η =   48 represents the outer computational boundary.

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Efficient Parallel Implementation of a Compact Higher-Order Maxwell Solver Using Spatial Filtering

Ramesh K. Agarwal , in Parallel Computational Fluid Dynamics 1999, 2000

6 RESULTS

As mentioned before, CEM code ANTHEM has been extensively validated by computing the TE and TM scattering from cylinders, airfoils and ogives by comparing the results with analytical solutions and MoM calculations. For the sake of brevity, here we present the results of parallelization of one case only, but similar performance has been achieved in other calculations. We consider TE scattering from a dielectric NACA0012 airfoil. The computational domain is shown in Figure 1. The hatching on the airfoil identifies it as a PEC NACA0012 profile. Each of the two zones wraps around the PEC airfoil forming a periodic continuous boundary at the x axis. The chordlength of the PEC airfoil is 4λ₀. The airfoil's thickness to length ratio is 0.12. A lossless dielectric coating of thickness t_coat = 0.1λ₀ surrounds the airfoil. The layer terminates as a blunt trailing edge having a thickness of 2 t_coat and extending t_coat beyond the trailing edge of the PEC airfoil. The radiation boundary has a diameter of 6λ₀. The inner zone represents the lossless dielectric layer between the PEC and dielectric interface. The lossless media is characterized by ∈ _c =7.4 and μ_c =1.4. The outer zone represents freespace.

Figure 1. Computational domain: (a) Zones, (b) Grid, every second line shown, zone 1:13 × l3, zone 2: 513 × 35.

The two-zone grid is shown in Figure 1(b). Due to the density of the grid, every other grid line is plotted. For the chosen dielectric constants, the speed of light is less than a third of that of freespace. Due to the reduced local incident wavelength and interference patterns, increased grid density is required in the dielectric zone.

Computations are performed for TE scattering from the dielectric airfoil due to an incident wave at 45° angle to the chord of the airfoil. Figure 2 shows the comparison of bistatic RCS computed with the CEM code ANTHEM and the MoM code. Figure 3 shows the scattered field D_z phase contours obtained with the CEM code. These calculations have been performed on a single HP750 workstation with both the explicit and implicit spatial filtering as shown in Table I. The implicit spatial filtering reduces the computational time by 20%.

Figure 2. Bistatic RCS: solid line, MoM solution, dashed line, ATHEM solution.

Figure 3. Scattered D_z field phase

Table I. Performance on 1 workstation – HP750

Single Zone =513 × 48 grid (513 × 13 in dielectric region)
	(a) Explicit Filter	(b) Implicit Filter
No. of Iterations *	9540	6580
Total cpu	4.568 h	3.510 h

Cpu/iteration/grid point: (a) 7 × 10⁻⁵ secs, (b) 7.8 × 10⁻⁵ sees.

*

Three-order of magnitude reduction in residuals

Tables I – III show the result of parallelization on one, two, and three workstations. Several domain decompostition strategies are implemented and evaluated on multiple workstations. In Table II, three strategies are implemented on two workstations. In strategy 1, two zones are divided at the dielectric/freespace interface; as a result there is load imbalance. In strategy 2, two zones are equally divided but the split is in the η-direction. In strategy 3, two zones are equally divided with a split in ξ-direction. Strategy 2 yields the best performance. On a cluster of three workstations, as shown in Table III, a similar conclusion is obtained. Again, the implicit spatial filtering reduces the computational time by 20 to 30%.

Table II. Performance on 2 workstations – 2 HP750

Domain Decomposition Strategy	Zone Dimensions		Number of Iterations		Parallel Efficiency
	Zone 1	Zone 2	Explicit Filter	Implicit Filter	Explicit Filter	Implicit Filter
1	513×14	513×36	9880	6840	91.9%	95.4%
2	513×25	513×25	9760	6750	93.2%	96.7%
3	258×48	258×48	10360	6930	87.6%	91.8%

Table III. Performance on 3 workstations – HP750, SGI Indigo, IBM RS/6000

Domain Decomposition Strategy	Zone Dimensions			Number of Iterations		Parallel Efficiency(based on clock time)
	Zone 1	Zone 2	Zone 3	Explicit Filter	Implicit Filter	Explicit Filter	Implicit Filter
1	513×17	513×17	513×17	9910	6740	87.2%	91.3%
2	172×48	172×48	172×48	10540	6870	82.0%	86.9%

Note:

ξ-grid line at η=1 represents the PEC NACA airfoil

ξ-grid line at η=13 represents the dielectric interface with freespace

ξ-grid line at η=48 represents the outer computational boundary.

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Noise From Rotating Machinery

William K. Blake , in Mechanics of Flow-Induced Sound and Vibration, Volume 2 (Second Edition), 2017

6.2 Elementary Acoustics of Rotating Machinery

6.2.1 Sources of Noise

Sounds from rotors, used either as single elements (as in propellers, fans, and helicopter rotors) or in combination with other fixed- or moving-vane arrangements in compound machinery (as in turbines, turbofans, etc.), may usefully be classified as interaction noise and self-noise. By interaction noise we mean all sounds that result from the encounter of a rotating blade with a time-varying disturbance in a frame of reference moving with the blade element. For purposes of clarity, it is well to note that two frames of reference are used in the analysis; one moving with the blades, the other moving with the axial flow machine. Recall, e.g., Fig. 5.2, which showed the time-varying loads and noise from a lifting surface as induced by an incident unsteady flow. By self-noise is meant sound resulting from flow over the blades themselves and requiring no unsteady inflow whatever. This latter source of rotor noise is generally due to viscous flow over the blades, whereas interaction noise is generally regarded as due to the potential reaction of the blades to local alternating angle of attack. The two types of noise probably are often mutually independent, although some changes in viscous flow can be effected by inflow unsteadiness. A form of self-noise arising from potential blade flow is so-called Gutin noise [29] (named for the first investigator who quantified its level, see Section 6.5.1), which occurs at multiples of the blade passage frequency, is proportional to the steady loading on the rotor, and is generally important at nearly sonic tip speeds and small blade numbers. This noise results from the forces on blade elements which are unsteady with respect to the acoustic medium because of the rotation of the blades, even though they are steady in a frame of reference moving with the blade. The measure of the acoustic radiation efficiency from this source is the ratio of the acoustic propagation time between blades to the rotation time between blades; this ratio is the rotational Mach number.

Important causes of interaction noise are

1.

rotor–stator interaction in compound turbomachinery;

2.

blade–blade tip vortex interaction (blade slap in helicopter acoustics) caused by flow across the rotor axis that forces a tip vortex of a forward blade to be overtaken by a following blade;

3.

inlet flow disturbances caused by secondary vortical flows and large-scale turbulence in forward stators, rotors, and grilles; and

4.

the interaction of rotor blades with annular boundary layers, as in ducted rotors.

The causes of self-noise are

1.

the continuous passage of boundary layer turbulence past trailing edges;

2.

laminar separation at any point on the rotor so as to cause unsteady blade pressures and, in an aggravated state, lift breakdown;

3.

laminar vortex development in the trailing wake;

4.

periodic vortex development in wakes of blunt trailing edges;

5.

rotation (Gutin) noise due to steady thrust and torque; and

6.

rotation monopole noise due to finite thickness of the blade section.

In all types of flow excitation, the elasticity of the blades could be important. Especially where trailing-edge vortex shedding is concerned (either of laminar or turbulent type), hydroelastic interaction gives rise to rotor blade singing, and all the principles of Chapter 5, Noncavitating Lifting Sections, apply. The resulting acoustic tone often appears at a fundamental frequency of the shedding, f _s (incidentally, f _s≫n _s B, where B is the blade number and n _s is the shaft rotation rate), with side bands spaced around f=f _s at intervals Δf equal to multiples of n _s B.

Fig. 6.1 shows many characteristics of rotor noise for the case of a helicopter rotor that is developing a time-varying thrust. The data is from Leverton [30], and the illustration is taken from Wright [6]. At low frequencies, tones arise at multiples m of the blade passage frequency (n _s B); these are identifiable in Fig. 6.1D up to the m=25 harmonic. Gutin sound from the rotation of the steady lift and torque vectors on the blades will occur at blade frequency and harmonics (f _m=mBn _s) and it has expected levels denoted by the envelopes labeled G. Observed noise levels in this frequency range exceed the expected Gutin rotational noise, especially at higher harmonics and low thrust. The excess noise, indicated by the scalloped envelopes, is due to transient interaction of the blade with the tip vortices of adjacent blades. This interaction is most prominent when there is no thrust and therefore little axial mean flow through the rotor disk. This axial flow would force the tip vortices into a helix (Fig. 1.3 shows the position of such vortices visible behind a cavitating marine propeller) away from the rotor. It should be noted that the helicopter rotor in this case was held fixed in a whirl tower, so the blade interactions resemble those of hovering rather than translating craft. The level and general frequency dependence of the scalloped envelope of the interaction noise is determined by the impulsive nature of the interaction, and the frequency interval between scallops is determined by the fraction of the circumference 2πR during which the impulse occurs. This type of noise will be the subject of Section 6.6.1.2.

Figure 6.1. Noise from a 16.7-m helicopter rotor on a whirl tower; measurement at 76.2   m distance, 11.5 degrees below the disk.Conditions A, B, C, D are identified below. Filter bandwidths: 2   Hz for 20–200   Hz; 5   Hz for 200–1500   Hz; 20   Hz for 1500–5000   Hz.

Adapted by Wright SE. The acoustic spectrum of axial flow machines. J Sound Vib 1976;45:165–223 from measurements of Leverton JW. The noise characteristics of a large "clean" rotor. J Sound Vib 1973; 27:357–76.

Two forms of broadband noise are apparent in Fig. 6.1. The broad hump centered on f=f _t occurs in many rotor signatures and the source of this noise has not been firmly established; it has been attributed both to turbulent flow over the blades [6,30–32] and interaction of blades with the turbulent wakes of other blades [14]. Noise in this frequency range has also [4,33] been attributed to the stalling of blade sections. Competing empirical correlations of this noise with the performance parameters of rotors for a wide variety of axial flow machinery [6] and for helicopters specifically [30] have shown a general increase in those broadband noise levels as either the total thrust, or the effective blade pitch, increases. This correlation would seem to be at least in line with Morfey's [33] suggestion that blade stalling, or some other form of turbulent boundary layer thickening on the blades, could enhance these levels (see Section 6.5.6). At much higher frequencies, near f _s in the figure, broadband noise and tones have both been observed. These noises are trailing-edge flow noises, much like those discussed in Chapter 5, Noncavitating Lifting Sections. Both sources of broadband noise will be discussed more fully in Sections 6.5.3 and 6.5.4.

6.2.2 Elementary Kinematics of Sound Radiation by Fan Rotors

Many important characteristics of the tonal quality of noises from rotating machinery can be determined simply from the kinematics of the blade interactions with the acoustic medium. Such arguments were applied in many of the earlier analyses of noise spectra (e.g., by Griffiths [34] and Cumpsty [35]). The general qualities of rotor–stator (or of rotor–rotor) interaction noises may be regarded as arising from circumferential spatial filtering in which samples of wake harmonics from the upstream components are made by the downstream blading system at integer multiples of the downstream blade number. By such filtering the downstream blading selectively responds to particular circumferential harmonics in its inflow that are generated upstream either by a wake-producing body or another blade row. The circumferential variations in the local inflow velocity may be called velocity "defects." The pertinent response of the blade row may consist of a summation of the force responses of the individual blade reactions at the fan shaft or it may consist of a resultant acoustic pressure in the far field due to the acoustically phased pressure contributions of individual blades. This behavior occurs actually done along the helix formed by the rotation and advance of the rotor through the fluid. For a temporally stationary mean inflow velocity defect, however, the velocity pattern is considered frozen over axial distances that are equivalent in magnitude to the rotor pitch so that a simple circumferential mode decomposition in the rotor plane is perfectly adequate to describe the circumferential harmonics of the inflow. Turbulent inflows, however, must be examined with regard to both circumferential and axial characteristics. We shall consider in this section only the generalities; the details will be developed in Section 6.6.

Solutions for the sound field of a complete distribution of rotor blade forces at finite rotational Mach number are analytically rather involved. Before delving into the complete problem, we shall consider a more idealized problem of a concentrated force generated by a blade of low pitch and moving at a vanishingly small Mach number. The summation process outlined above will thus now be examined for an elementary case, but the result will have a general form. The analysis will disclose features in the frequency spectrum of radiated sound that are typical of low-speed axial flow fans. These features can be addressed simply by expanding Eq. (2.73) of Volume 1 for the case of a rotating distribution of sources. The appropriate geometry is given in Fig. 6.2. We assume a force increment at the location (R, θ _s, y _s=0) with axial and tangential force components f ₁ and f _θ , respectively. These forces are related to the resultant force f, and the tangential force f _θ may be further resolved into components f ₂ and f ₃ in the rotor plane. Thus the constituent forces are given by

Figure 6.2. Geometric projections and blade indexing rotor blade systems. (A) Projections of source and field points for rotating systems and (B) blade indexing for rotor.

$\begin{array}{l}{f}_1=f\cos \gamma \hfill \\ f_2=-f\sin \gamma {\sin \theta }_{\mathrm{s}}\hfill \\ f_3=f\sin \gamma {\cos \theta }_{\mathrm{s}}\hfill \end{array}$

where θ _s is the angle of the source in the (R,θ), or (2,3) plane, R is the rotation radius, and γ is the angle of the force measured in a plane that is perpendicular to the R axis; it is equivalent to the hydrodynamic pitch angle in marine technology and the complement of the stagger angle in compressor design.

The source–observer vector r _s may also be resolved into its constituent components:

$\begin{array}{l}{(r_{\mathrm{s}})}_1=r\cos \beta \hfill \\ {(r_{\mathrm{s}})}_2=r\sin \beta \cos \theta -R\cos {\theta }_{\mathrm{s}}\hfill \\ {(r_{\mathrm{s}})}_3=r\sin \beta \sin \theta -R\sin {\theta }_{\mathrm{s}}\hfill \end{array}$

r _s is instantaneously variable as θ _s varies with the rotation of the source. The generation of tones and their harmonics is determined by this rotation. In the analysis, it is important to express the variable r _s in terms of a fixed radius r measured from the axis of rotation. It is assumed that r≫R, and from the geometry the magnitude of r _s is

(6.1) $r_{\mathrm{s}}=\sqrt{r^2+R^2-2Rr\sin \beta \cos (\theta -{\theta }_{\mathrm{s}})}$

which can be approximated by

(6.2) $r_{\mathrm{s}}\simeq r-R\sin \beta \cos (\theta -{\theta }_2)$

when r ≫ R. Note that the source angle θ _s is time varying; i.e.,

(6.3) ${\theta }_{\mathrm{s}}={\theta }_0+\mathrm{\Omega }t$

where Ω is the rotation rate of the blade. In the simplified problem of this section we assume that the force f is spatially concentrated, but it is not necessary that the rotation radius R be small compared to an acoustic wavelength; i.e., k ₀ R remains arbitrary. Eq. (2.73) of Volume 1 may then be written

(6.4a) $p_{\mathrm{a}}(\mathit{x},t)=-\frac{1}{4\pi }\frac{\partial }{\partial x_i}\left[\frac{f_i(t-r_{\mathrm{s}}/c_0)}{r_{\mathrm{s}}}\right]$

where differentiation is with respect to the field coordinate in the direction of the force. Substituting for the force components, this expression expands to give the far-field pressure by carrying out the differentiation such as that leading to Eq. (2.73) of Volume 1

$\frac{\partial }{\partial x_i}\left(\frac{f_i}{r_{\mathrm{s}}}\right)\approx \frac{-1}{c_0}\left(\frac{1}{r}\frac{\partial f_i}{\partial t}\right)\frac{\partial r_{\mathrm{s}}}{\partial x_i}$

for which

$\frac{\partial r_{\mathrm{s}}}{\partial x_i}=\frac{r_{{\mathrm{s}}_i}}{r_{\mathrm{s}}}$

Thus,

$p_{\mathrm{a}}(\mathit{x},t)=\frac{1}{4\pi c_0}\left\{\frac{\dot{f}(t-r_{\mathrm{s}}/c_0)}{r_{\mathrm{s}}}·\left(\frac{r_{{\mathrm{s}}_1}}{r_{\mathrm{s}}}\cos \gamma +\frac{r_{{\mathrm{s}}_2}}{r_{\mathrm{s}}}\sin \gamma \sin {\theta }_{\mathrm{s}}+\frac{r_{{\mathrm{s}}_3}}{r_{\mathrm{s}}}\sin \gamma \cos {\theta }_{\mathrm{s}}\right)\right\}$

or the pressure due to this force increment is

(6.4b) $p_{\mathrm{a}}(\mathit{x},t)=\frac{1}{4\pi c_0}\left\{\frac{\dot{f}(t-r_{\mathrm{s}}/c_0)}{r_{\mathrm{s}}}·(\cos \beta \cos \gamma +\sin \gamma \sin \beta \sin (\theta -{\theta }_{\mathrm{s}})\right\}$

for r≫R. The instantaneous angle θ _s is

${\theta }_{\mathrm{s}}={\theta }_0+\mathrm{\Omega }t$

where θ ₀ is a reference angle and Ω is the rotation angular velocity, for now, retardation effects will be ignored in the instantaneous angle. In succeeding sections of this chapter, we will consider various ways of analytically describing the sound fields of various types of force fields. Our current interest is to examine some of the kinematic features of low-speed rotating blades and this is most clearly done by emphasizing the axial forces only. The result becomes exact for small blade pitch and for angles β which are sufficiently removed from the propeller plane, i.e., for

$0\le \beta <\frac{\pi }{2}-\frac{R}{r}$

The blade force increment then reduces to f(t)=f ₁(t), which is directed axially, r parallel to the one axis.

To continue examining the basics, we shall assume that the force is harmonic at the frequency ω _s in the blade's frame of reference, i.e., that

(6.5) ${\dot{f}}_1(t-r_{\mathrm{s}}/c_0)=-i\omega F_1({\omega }_{\mathrm{s}})e^{-i({\omega }_{\mathrm{s}}t-k_0{r}_{\mathrm{s}})}$

a more general time dependence will be the subject of Section 6.4. The bracketed term in Eq. (6.4a) is thus

(6.6) $\frac{f_1(t-r_{\mathrm{s}}/c_0)}{r_{\mathrm{s}}};\frac{F_1({\omega }_{\mathrm{s}})e^{-i({\omega }_{\mathrm{s}}t-k_{0}r)}}{r}{e}^{-i{k}_{0}R\sin \beta \cos (\theta -{\theta }_{\mathrm{s}})}$

for r≫R. It is useful at this point to introduce a summation formula [36]

(6.7) $e^{-i{k}_{0}R\sin \beta \cos (\theta -{\theta }_{\mathrm{s}})}=\sum _{n=-\infty }^{\infty }{(-i)}^n{e}^{in(\theta -{\theta }_{\mathrm{s}})}{J}_n(k_{0}R\sin \beta )$

where J _n (x) is the cylindrical Bessel function [36] of nth order, further description of which will be given in Section 6.4. Substitution into Eq. (6.6) gives

(6.8) $\frac{f_1(t-r_{\mathrm{s}}/c_0)}{r_{\mathrm{s}}}=\frac{1}{r}\sum _{n=-\infty }^{\infty }{(-i)}^n{e}^{-i({\omega }_s+n\mathrm{\Omega })t}{F}_1({\omega }_{\mathrm{s}})e^{i(k_{0}r+n(\theta -{\theta }_0))}{J}_n(k_{0}R\sin \beta )$

This equation shows that, for a rotating tonal source with frequency ω _s, the radiated sound has an infinite number of side bands at intervals ±nΩ on either side of this frequency. This behavior is due to Doppler shifting of the primary frequency, and the occurrence of an infinite set of harmonics is due to a periodic Doppler shift.

Thus repeating the far-field differentiation procedure, we obtain

(6.9) $p_{\mathrm{a}}(\mathit{x},t)=\frac{1}{4\pi }\frac{\cos \beta }{r}\sum _{n=-\infty }^{\infty }{(-i)}^{n+1}{k}_0{F}_1({\omega }_{\mathrm{s}})e^{-i({\omega }_{\mathbf{s}}+n\mathrm{\Omega })t}{e}^{i(k_{0}r+n\theta )}{J}_n(k_{0}R\sin \beta ).$

which is just a restatement of the first term of Eq. (6.4b), but now tailored to our cylindrical coordinate system.

Recall that in Section 4.6 the treatment of the sound field of a rotating rod ignored these rotational effects. The generation of these side bands on either side of a predominant frequency ω _s may be ignored whenever

1.

sin β=0, i.e., on the axis, since J _n (0)=0 for n≠0 and J ₀(0)=1 or

2.

when any distribution in energy over the frequencies Δω>Ω overshadow the side bands.

In the case of the rotating rod, the source was not localized to the tip, but rather was distributed in magnitude and frequency along the radius of the rod. The continuous dustribution manifested itself as a spectrum without evidence of side bands.

6.2.3 Features of Sound From Inhomogeneous Inflow

We continue our survey of the elementary aspects of rotation sound by now considering compact forces generated by the interaction of a blade with a time-invariant spatial harmonic of inflow variation. We will forgo discussion of the specific fluid dynamic causes of the forces until Sections 6.5 and 6.6, but in this subsection we will assume that the frequency dependence of forces on the blades is the result of the blade's encounter with an inflow spatial wave form of the distorted mean flow and that these distortions in velocity are convected across the blade's chord without modification. Thus the blade chord is considered small in comparison with an acoustic wavelength, but the rotor diameter is not. Also, the blade loading is currently considered to be concentrated over a radial extent which is also acoustically small. Assume an inflow to the rotor of circumferential harmonic character, say,

(6.10) $u(\theta )=U_{\mathrm{w}}{e}^{iw\theta }$

where w is an integer. The rotating blade encounters this velocity fluctuation at a rate

$\frac{d\theta }{dt}=\mathrm{\Omega }$

Assume further that, as shown in Fig. 6.2B, that there are B blades equispaced at angles 2π/B around the axis of the rotor. Accordingly, the velocity u(θ) at the sth blade is

$u({\theta }_{\mathrm{s}})=u\left({\theta }_0+s\left(\frac{2\pi }{B}\right)-\mathrm{\Omega }t\right)$

As discussed in Chapter 5, Noncavitating Lifting Sections, the force response of each blade to this velocity fluctuation is dependent on the encounter frequency, the blade chord, the aspect ratio of the blade, and the spanwise uniformity of the blade loading. Regardless of the details of the blade response, the sth blade force from the wth inflow harmonic will be of the form

(6.11) ${[F(t)]}_{s,w}={\left|F\right|}_{s,w}{e}^{iw({\theta }_0+s(2\pi )/B-\mathrm{\Omega }t)}$

with a frequency of encounter wΩ=ω _s. For simplicity in the current analysis, we shall still assume that this force acts axially, i.e., F=F ₁. The factor s(2π/B) simply indexes each sth blade around the disk of the rotor. Now, there are three indexes of summation: the index n, which pertains to the acoustic phasing due to motion of the blades; the index 0<s<B−1   used to sum the contributions from all the blades; and the index w for the harmonics that the inflow imposes on the rotor. Thus Eqs. (6.6), (6.7), and (6.11) combine to yield for a single inflow harmonic, w,

(6.12) $\begin{array}{ll}{\left({\displaystyle \frac{f_1(t-r_{\mathbf{s}}/c_0)}{r_{\mathbf{s}}}}\right)}_w\simeq \hfill & \sum _{s=0}^{B-1}\sum _{n=-\infty }^{\infty }\frac{{\left|F_1\right|}_{s,\omega }}{r}{e}^{iw({\theta }_0+s2\pi /B-\mathrm{\Omega }t)}\hfill \\ \hfill & \times {(-i)}^n{e}^{i{k}_{0}r}{e}^{in({\theta }_0-\theta +s2\pi /B-\mathrm{\Omega }t)}{J}_n(k_{0}R\sin \beta )\hfill \end{array}$

This rather complicated-looking pair of summations is simplified when it is noted that it involves two terms:

$\begin{array}{lll}{e}^{+i(w+n)(2\pi /B)s}\hfill & \mathrm{and}\hfill & e^{-i(w+n)\mathrm{\Omega }t}\hfill \end{array}$

The summation over all B blades is

(6.13) $\sum _{s=0}^{B-1}{e}^{iw(2\pi /B)s}=\sum _{m=-\infty }^{\infty }B\delta (w-mB)$

because it is a partial summation over $e^{+iw(2\pi /B)s}$ , which is a geometric progression, $a_0+a_0^2\cdots +a_0^{\mathrm{s}}$ . Eq. (6.12) simplifies to give the radiated sound pressure as

(6.14) $\begin{array}{ll}{p}_{\mathrm{a}}(\mathit{x},t)=\hfill & \sum _{m=-\infty }^{\infty }\frac{B}{4\pi }\frac{{\left|F_1\right|}_w}{r}{k}_0\cos \varphi e^{-imB\mathrm{\Omega }t}{e}^{i{k}_{0}r}{(-i)}^{mB-w+1}\hfill \\ \hfill & \times e^{i(mB-w)\theta }{J}_{mB-w}\left(\frac{mB\mathrm{\Omega }R}{c_0}\sin \beta \right)\hfill \end{array}$

where the summation over s yields Bδ(w+n−mB) and k ₀=mBΩ/c ₀. It has been assumed that the amplitudes of unsteady loading on each blade are the same, making |F ₁| _s,w =|F ₁| _w . The form of this equation is fundamental to all source mechanism discussed in this chapter. From this point on, we will examine the sounds from various types of harmonic and anharmonic distortion fields.

This equation shows a number of important characteristics of the sound from the axially directed force induced by inflow harmonics:

1.

The sound is emitted at frequencies that are multiples of the blade passage frequency ω=BΩ. For each harmonic w the strongest harmonic is at mB=w, and this harmonic becomes more important than the others as (mBU _T/c ₀) sin β→0 at low-tip-speed Mach numbers, where U _T=ΩR.

2.

For higher-order m, so that mBM _T sin β is nonnegligible, additional radiating modes occur, and these modes propagate outward along a constant phase trajectory that spins at (mB±w)θ=mBΩt, or at angular velocity

$\frac{\partial \theta }{\partial t}=\left(\frac{mB}{mB\pm w}\right)\mathrm{\Omega }$

These are called spinning modes, and they are not apparent at β=0. These modes will be discussed further in Section 6.6.1.1.

3.

At low-tip-speed Mach numbers, for which J _n (mBM _T sin β) ≈ 0 for n ≠ 0, the only inflow harmonics w that radiate sound (or, equivalently, generate a net time-varying axial force on the fluid) are those for which mB=w. Thus the rotor of B blades responds principally to those inflow harmonics that are multiples of the number of rotor blades.

4.

For uniform inflow, w=0, radiated sound also exists, but Eq. (6.14) shows that its intensity is predominant at high tip speed. At Mach numbers such that J _mB (mBM _Tsinβ) are nonnegligible, such sound is called Gutin sound [29].

In the frame of reference rotating with the rotor blades, the inflow is thus viewed as spinning at the rotational speed of the rotor. The response of each blade to the unsteadiness is given by the airfoil admittance function times the intensity of the incident disturbance, as discussed in Chapter 5, Noncavitating Lifting Sections. The discrete spatial filtering comes about from the scanning of the inflow by the blade array, as illustrated in Fig. 6.2. At low-tip Mach numbers, the acoustic signal emitted is then proportional to a sum of the responses of the individual blades, with time delay (or relative phases) given by the ratio of the blade spacing to the wavelength of the sound emitted. At low enough Mach numbers that the wavelength of the sound is larger than the rotor circumference, the rotor acts simply as a summing device that gives the resultant sound proportional to the resultant force components (thrust and torque) induced on the rotor. In this case a maximum summed response of the rotor occurs for all circumferential wavelengths that are integral numbers of the blade spacing. This can be seen by noting that the resultant axial force is given by Eqs. (6.12) and (6.13) by a summation over all harmonics m of the blade passage frequency

(6.15) ${(F_1)}_w=B\sum _{m=-\infty }^{\infty }{(F_1)}_{s,w=mB}{e}^{-imB\mathrm{\Omega }t}$

and (F ₁) _s,w=mB is the force on any blade that has a particular amplitude and phase depending on wake harmonic w.

An analog to these results is in the context of the spatial filter, as discussed in Section 2.5.1. The rotor as a continuous distribution of evenly-spaced blade responses acts as a circumferential spatial filter. The circumferential wave number of the inflow velocity defects are k _θ =w/R, where w is an integer and the rotor blades respond in phase to wave numbers k _θ =mB/R with m=0, 1, …. The effect of the rotation of the blades is to translate nonuniformities in θ into temporal nonuniformities such that the frequency of blade encounter with the w/R wave vector component is

$\omega =k_{\theta }{U}_{\theta }=\left(\frac{w}{R}\right)\mathrm{\Omega }R=w\mathrm{\Omega }.$

Fig. 6.3 illustrates this behavior in various ways for a rotor of eight blades in an inflow of V cycles where V=2B=16. The circumferential acceptance wave numbers are shown in Fig. 6.3A. The response of the rotor is concentrated at circumferential harmonics that are multiples of the number of inflow cycles, in this case V. The response is tapered by the Sears function which is a function of both chord and circumferential order. Fig. 6.3B shows the characteristics of the inflow; because of the rotation, the spectral content of the inflow lies on a frequency-harmonic, line as shown. Fig. 6.3B also has illustrated on its right hand side a random component, which is a continuous spectrum in the circumferential wave number; i.e., it is nonharmonic. This line represents the continuous spectrum of turbulence in the inflow. The discrete harmonics occurring at k _θ R=w=V, 2V, 3V, … are the harmonic orders of the mean flow distortion. The small sketch on the upper right of Fig. 6.3B illustrates the random turbulence on the time-mean distortion plotted over a period 2πR/V Fig. 6.3C represents a frequency spectrum of the resultant rotor thrust fluctuations that is formed as a product of Fig. 6.3A and B, summed over harmonic order, and plotted versus frequency-to-shaft speed ratio that have harmonics at all mB=nV, i.e., at 2B=V, 4B=2V, etc. Harmonics are not present for ω=BΩ, 3BΩ, …, because there are no w harmonics that are coincident with rotor acceptance wave numbers. At these frequencies, lower level humps appear owing to turbulence–blade interactions and having bandwidths determined by both the statistics of the inflow turbulence and the blade response. These features will be discussed in Section 6.6.2.

Figure 6.3. Interpretation of a rotor response to inflow nonuniformity as a filter responding to a signal plus noise. Turbulence acts as noise because it is broadband and because its macroscale is larger than blade spacing. (A) Components of load response as a function of circumferential order, sketch of the rotor on the right; (B) Characteristics of the energy density of inflow as functions of circumferential order, with representations of the inflow and its circumferential order spectrum on the right; and (C) the net force response of the blade row as a spatial array.

For forces directed parallel to the axis of the propeller, the acoustic radiation is given by equations of the type of Eq. (6.9) or (6.14), depending on the physical nature of the rotating force. Quantification of the sound, beyond the general description of the spectrum shape presented here, depends on the forces predicted and the specific functional form of (F ₁) _s,w as to be discussed in Section 6.6. Furthermore, a more sophisticated analytical modeling of the force-generation mechanism will also account for its distribution along the radius of the rotor although the blade's dipoles are compressed to the plane of the rotor. The remainder of this chapter will treat the rotor blades as lifting surfaces and analytical methods that were developed in Chapter 5, Noncavitating Lifting Sections.

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Introduction to the Problems of Flow Separation

PAUL K. CHANG , in Separation of Flow, 1970

4.2 Crocco–Lees' mixing theory

A simplified form of the mixing theory was developed, as mentioned previously, by Crocco and Lees [8] and is applicable not only for separated and reattached flows but also for wake flow to yield qualitative results. By this method of analysis, a qualitative agreement is reached between the results of theoretical calculations of base pressure versus Reynolds number and the experimental data of Chapman [52] and Bogdonoff [53] on bodies of revolution, and Chapman's data [54] on blunt trailing-edge airfoils. Thus this is more widely applied for base pressure problems, although Crocco–Lees' theory is a general solution of separated flow. It has been found that separated as well as reattached flow are capable of supporting considerable pressure increase at high velocities. Until the Crocco–Lees' theory was developed, assumption of a uniform static pressure in the wake and jet prevailed in the analysis of viscous flow. In reality, this simple assumption does not hold. Crocco and Lees found that in separated flow the pressure gradient along the surface may reach a maximum at separation and may drop off steeply downstream, and that in reattaching of wake flows the pressure gradient is negligible some distance upstream of the reattachment point and increases rapidly as this point is approached. In flow separations two governing parameters x and f are defined on the basis of the boundary layer as

$\begin{array}{lll}x=\left(\delta -{\delta }^{*}-\theta \right)/\left(\delta -{\delta }^{*}\right)\hfill & and\hfill & f=\left({\delta }_i-{\delta }_i^{*}-{\theta }_i\right)·{\delta }_i/{\left({\delta }_i-{\delta }_i^{*}\right)}^2\hfill \end{array},$

where δ, δ*, and θ are physical, displacement, and momentum thicknesses of boundary layer respectively and the subscript i denotes the incompressible flow in the Stewartson transformation.

For laminar and turbulent flows, the separation is characterized by decreasing x and by increasing f. At the separation point, the slope ∂f/∂x is infinite or vertical, as shown in Figs. 50 and 51. This fact has a clear physical interpretation.

FIG. 1.50. Laminar flows [8]

FIG. 1.51. Turbulent flows [8]

With approaching separation, the effect of the wall on the high-velocity portion becomes less and less marked, and the conditions in the portion which determines the values of x become more and more similar to those of a "free" half-jet with a constant x-value. On the other hand, f increases steadily because of the increase in total thickness of the dissipative flow. Hence ∂f/∂x becomes infinitely large at the separation point.

This physical picture is essential for separated flow solutions since it gives a clue to the qualitative prediction of what happens after separation and because no reliable theoretical solution has been established in that region.

It is to be expected that in the separated region the jet-like behavior is still more pronounced; therefore x undergoes only small variations, while all the variations in the flow will be reflected in f. Consequently, the f − x-curve will have a nearly vertical extension beyond the separation point.

The reattachment of the separated flow to the wall and the subsequent build-up of a regular boundary layer again can be predicted in the f − x-plane analogous to what is observed for separating and separated flows, but in the opposite direction.

In the case of wake far behind a body, ${\delta }_i^{*}=\theta =0$ and x = f = 1. Therefore it may be stated that every incompressible dissipative flow involving separation, separated, reattached regions, and wakes can be described by a particular curve in the f – x-plane. If Stewartson's transformations for turbulent flow are assumed to be applicable for separated flow, the compressible dissipative flow can be described by the same f – x relationship. Thus Crocco–Lees' theory, although its solution is qualitative, is a powerful tool to solve steady fluid mechanics problems involving separation.

This analysis is applicable not only for two-dimensional flow but also for axisymmetric subsonic as well as supersonic flows. Recently, Glick [36] found that the assumption of an improper f(x) relation caused the discrepancy between Crocco–Lees' theory and experimental data in the flow region up to separation, and modified Crocco–Lees' theory.

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Noncavitating Lifting Sections

William K. Blake , in Mechanics of Flow-Induced Sound and Vibration, Volume 2 (Second Edition), 2017

5.2.2.3 General Applicability of Eq. (5.6): The Kutta Condition

Eq. (5.6) can also be interpreted as the pressure at (r ₀, ϕ ₀, δ ₀) due to a source at (r, ϕ, θ). On the rigid plane surface in the region y ₁<0 (i.e., r ₀ sin ϕ ₀>0 and θ ₀=±π), the normal velocity of the fluid, expressed by ∂G/∂y ₂, is zero. This velocity in the wake (θ ₀=0) becomes singular, however, as $\underset{1}{\overset{-1/2}{y}}$ (or as (r ₀ sin ϕ ₀)^−1/2) as the edge is approached, i.e., as y ₁→0. This singularity in ∂G/∂y ₂ as y ₁→0 implies that the pressure differential between the upper and lower surface also becomes singular. When a Kutta condition is applied in the analysis, this singularity is removed by adding functions that exactly cancel the growing singularity as y ₁→0. Accordingly the Kutta condition can be provides for the shedding of vorticity downstream of the edge into the wake so that a physically reasonable pressure differential is maintained at the edge. The magnitude of this vorticity is determined by the requirement that the pressure differential induced across the surface by the wake is just sufficient to exactly cancel the singular pressure differential induced by the approaching upstream vortex. The flow thus "wets" the edge and responds to its presence. The shed vorticity in the wake is convected at a velocity U _W. The distinction between boundary conditions is shown schematically in Fig. 5.6. When no Kutta condition is applied in analysis, a singular pressure is allowed at the edge and the pressures on opposite sides are π out of phase. When the Kutta condition is applied, the singularity is removed. A "complete" Kutta, or Kutta–Joukowski, condition removes the differential pressure completely. The measured pressures, discussed in Sections 5.3 and 5.4, show that the differential in surface pressure between the upper and lower surfaces increases as $y_1^{-1/2}$ in two physical circumstances: at the leading edge of a lifting surface responding to upstream flow inhomogeneities (so-called leading edge noise) and at blunt trailing edges downstream of which a vortex street is formed in the wake. These results suggest that no Kutta condition should be applied in pertinent analyses. Such apparent singularities are not formed on the surface of wall jets for which the upstream boundary flow is turbulent and on some sharp-edged airfoils at high Reynolds number, indicating that a Kutta condition is appropriate in these cases. Thus the application of Kutta conditions must be done with care, and one mathematical form may not be universally valid for all types of flows.

Figure 5.6. Behaviors of surface pressures induced by vortex–edge interactions and associated with alternative pressure boundary conditions at edges. The case shown is a one-sided wall-flow passing an edge such as a blown flap or wall jet. Pressures p _u and p _ℓ are referred to a value of 2 on the wetted side far from the edge. (A) No "Kutta" condition and (B) "Kutta" condition.

Howe [12] has examined the implications of a mathematical trailing edge Kutta condition for the two-dimensional problem of an infinitely long vortex filament with axis parallel to the edge y ₃ and approaching the trailing edge from the surface side (y ₁<0) by moving in the y ₁ direction at a velocity U _c. The geometry is illustrated in Fig. 5.6. When a trailing edge singularity is permitted in the analysis, i.e., for no Kutta condition, the radiated pressure is given by

(5.12) $p_{\mathrm{a}}(\mathit{x},t)=\frac{{\rho }_0{\mathrm{\Gamma }}_3{U}_{\mathrm{c}}\sin (\theta /2)}{2\pi \sqrt{r}}{\left[\frac{\cos {(\theta }_0/2)}{r_0^{1/2}}\right]}_{t-r/C_0}$

where the bracket denotes the location of the vortex is that the earlier time t−r/c ₀ and where Γ₃ is the circulation of the vortex. This form closely resembles the three-dimensional result that would result from substituting Eq. (5.8b) into Eq. (5.10), but it has the $1/\sqrt{r}$ geometrical spreading loss that is characteristic of two-dimensional acoustics problems (see the Appendix, Chapter 4 of Volume 1). When a complete Kutta condition is applied, i.e., when the pressure differential at y ₁=0 is taken to be zero, the sound pressure emitted is reduced from the case of no Kutta condition by an amount (1−U _w/U _c) because of the required shed vorticity. Thus the sound pressure in the case of a Kutta condition applied is given by

(5.13) $p_{\mathrm{a}}(\mathit{x},t)=\frac{{\rho }_0{\mathrm{\Gamma }}_3{U}_{\mathrm{c}}\sin (\theta /2)}{2\pi \sqrt{r}}\left(1-\frac{U_{\mathrm{w}}}{U_{\mathrm{c}}}\right){\left[\frac{\cos {(\theta }_0/2)}{r_0^{1/2}}\right]}_{t-r/C_0}$

Therefore, if the convection velocity U _c is equal to U _w, the radiated sound is identically zero. Observations by Yu and Tam [11] of the vortex structure in a wall jet disclose that, in response to upstream eddies, wake eddies are shed at a velocity U _w≈0.6U _c. This result suggests that a Kutta condition should apply to such flows involving upstream boundary layer turbulence and that the magnitude of sound pressure radiated could be significantly less than the value given by theories based on classical acoustic diffraction such as that used in Section 5.2.2.2. The functional behavior, though, is the same for both boundary conditions.

In summary, these results, like the results of measurements to be described in Sections 5.3–5.6, suggest that edge boundary conditions that permit a $1/\sqrt{y_1}$ dependence of the differential surface pressure as y ₁→0 apply to leading edge noise and to vortex shedding noise. The singularity must be removed by the application of a Kutta condition for those cases involving wall jets, blown flaps, and the upstream boundary layer turbulence convected past the edge. Essentially, this condition amounts to the requirement that the flow leave the edge tangentially with respect to both the mean and the instantaneous velocities in the immediate vicinity of the edge. For either leading or trailing edge flows, the essential dependence of the sound on the flow parameters is still that given by Eqs. (5.11a and b) but with differing coefficients of proportionality.

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Application of Constrained Target Pressure Specification to Takanashi's Inverse Design Method

T.E. Pambagjo , ... S. Obayashi , in Inverse Problems in Engineering Mechanics III, 2002

DESIGN RESULTS

The goal of this study is to investigate the use of constrained target pressure specification method described in above section to design a thick airfoil and a wing for BWB airplane.

The flowfield solution is obtained by using the Navier-Stokes equations. The boundary layer on the airfoil was assumed to be fully turbulent and the Baldwin-Lomax turbulence model is used. The Navier-Stokes solver [4] adopted the LU-SGS method for the time integration and the 3rd order upwind scheme for the space discretization of the convective terms.

Case 1

The objective is to design an airfoil with a maximum thickness ratio of 12%. The aerodynamic constraints include a lift coefficient of 0.6, a pitching moment coefficient greater than −   0.15 and wave drag coefficent less than 0.0005. The flow condition is at a free stream Mach number of 0.75, an angle of attack at 0   deg., and a Reynolds number of 1.0   ×   10⁷. The Navier-Stokes equations are solved using C type of mesh which contains 161   ×   93 grid points.

The initial target pressure distribution is generated by utilizing the empirical estimation approach as to meet the constraints. After the design location of the shock wave is determined during the iteration, the automated modification approach is employed. At this stage the target pressure is regenerated based on the intermediate pressure distribution. After the control points are obtained then the shape of the pressure distribution of the initial one is used to regenerate the target pressure distribution between the control points.

From this stage after every five cycle the target pressure is modified until the result converges to the constraints. The design result is a maximum thickness ratio of 12%, a lift coefficient of 0.6, a wave drag coefficient of 0.00012 and a pitching moment coefficient of −   0.1455. The total drag coefficient is 0.0127. These results satisfy the specified constraints successfully.

Figure 4 shows the design result. The final target pressure is achieved except in the shock wave area.

Fig. 4. Design Airfoil shape and corresponding pressure distribution for case 1

Case 2

The objective of case 2 is to design a thicker airfoil. The desired maximum thickness ratio is 18% with aerodynamic constraints which are a lift coefficient of 0.4, a pitching moment coefficient greater than −   0.11. The flow condition is set at a free stream Mach number of 0.74, an angle of attack at 0   deg., and a Reynolds number of 10   ×   10⁶. NACA 0012 is used as the initial geometry for the inverse design process. The Navier-Stokes equations are solved using C type of mesh containing 161   ×   93 grid points.

In this case the initial target pressure distribution is specified by utilizing the control point fitting approach. The pressure distribution of NASA supercritical airfoil [5 ], NASA SC(2)-0518, is used as reference for generating the target pressure distribution. Although this airfoil originally has a blunt trailing edge, for the convenience the airfoil is modified to have a sharp trailing edge. This does not violate the design concept. With the flow condition, the aerodynamic properties of NASA SC(2)-0518 are a lift coefficient of 0.38, a total drag coefficient of 0.0191 and a pitching moment coefficient of −  0.1134. The wave drag is 0.0014.

After the target shock wave location is captured, the process is continued by utilizing the automated target generation. At the final design, the properties of design airfoil are a lift coefficient of 0.401, a wave drag coefficient of 0.00012 and a pitching moment coefficient of −   0.1089. The maximum thickness ratio is 18%. The total drag coefficient is 0.0148. These properties indeed meet with both the flow and geometric constraints.

Figure 5 shows the design result. The design airfoil has more thickness in the aft portion and in leading edge area. Although in general the target pressure is achieved, there is still small discrepancy in the shock area.

Fig. 5. Designed Airfoil shape and corresponding pressure distribution for case 2

Case 3

The BWB airplane is a conceptual airplane, which in essence is a flying wing. In this configuration the passengers are accommodated in the inboard wing. This will require thick airfoil in the inboard wing. Figure 6 shows the configuration, which is used in this study.

Fig. 6. Design configuration

Initially the target pressure distributions are developed using the control point fitting approach and during the design process the target pressure distributions are modified iteratively to meet the constraints. NASA supercritical airfoils [5] are chosen as the initial airfoil for the outboard section, for the inboard section 4 digit NACA series airfoils are used. The geometry constraint here is limited to the maximum thickness requirement at each design locations, which are determined based on the space requirement. The other goal is to have lower drag than the initial wing by reducing the wave drag. At this phase constrained target pressure specification technique [2] is very useful. It helps constructing the required target pressure distribution that leads to the desired requirements. However, usually manual adjustment is still required.

The author would like to focus on the result of the inboard wing design because inboard wing design is much more difficult than outboard wing. The inboard wing is created using four wing sections at several design locations. Those design locations are at 0%, 12%, 24% and 40% semi span. At those locations the wing sections are obtained by using Takanashi's inverse design method, then the RAPID (Rapid Airplane Parametric Input Design) method [6] generates the wing surfaces. RAPID method generates smooth surface by solving the fourth order differential equation. The target pressure distribution has been defined based on the design requirement. The aim here is to obtain the thickness distribution which lead to the geometry requirement of the inboard wing, which can carry the payload. The other aim is to have drag and pitching moment coefficient as small as possible.

To evaluate the aerodynamic performance of the wing, the Navier-Stokes equations were solved using C-H type mesh contains 191   ×   50   ×   49 grid points. The flow conditions is set at free stream Mach number of 0.8 and the angle of attack of 0   deg., and the Reynolds Number is 10⁷.

Figure 7 shows the results of the inverse design process at several design locations. It shows that the design processes converge to the specified target pressure distribution. However there are small differences near the leading edge and trailing edge. Table 1 shows the aerodynamic performance of the design compare to the initial wing. The drag coefficient is lower than the initial one; this might come from a result that the thinner airfoil in the inboard section has been designed. However the pitching moment (reference point is leading edge of the center airfoil) is slightly higher than the initial one because of the higher aft loading in the inboard section. It can be improved if the target pressure distribution is well adjusted for aerodynamic forces to desirable feature.

Fig. 7. Inverse design results of the BWB wing design.

Table 1. Aerodynamic performance

	Design	Initial
Lift coefficient, CL	.5124	.5136
Drag coefficient, CD	.0272	.0312
Pitching moment coefficient, Cm	-.3563	-.3353
Lift to Drag ratio, L/D	18.87	16.463

Figure 8 shows the airfoils at the three design locations in the fuselage. The passenger's cabin is set inside those three airfoils. In this figure the passenger's cabin is also represented as the rectangle. The figure shows that the present method can achieve the aerodynamic target with satisfying the geometry constrains to obtain wide space for the passenger's cabin.

Fig. 8. Passenger's cabin

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Source: https://www.sciencedirect.com/topics/engineering/blunt-trailing-edge

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