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Flow around the front pillar of an automobile is typical of a flow field with separated and reattached flow by a vortex system. It is known that the vortex system causes the greatest aerodynamic sound around a vehicle. The objective of the present study is to clarify the relationship between vortical structures and aerodynamic sound by the vortex system generated around the front pillar. The vortex system consists of the longitudinal and the transverse system. The characteristics of the longitudinal vortex system were investigated in comparison with the transverse one. Two vortex systems were reproduced by three-dimensional delta wings. The flow visualization experiment and the computational fluid dynamics (CFD) captured well the characteristics of the flow structure of the two vortex systems. These results showed that the longitudinal with the rotating axis along mean flow direction had cone-shaped configuration whereas the transverse with the rotating axis vertical to mean flow direction had elliptic one. Increasing the tip angles of the wings from 40 to 140 degrees, there first exists the longitudinal vortex system less than 110 degrees, with the transition region ranging from 110 to 120 degrees, and finally over 120 degrees the transverse appears. The characteristics of aerodynamic sound radiated from the two vortex systems were investigated in low Mach numbers, high Reynolds number turbulent flows in the lownoise wind tunnel. As a result, it was found that the aerodynamic sound radiated from both the longitudinal and the transverse vortex system was proportional to the fifth from sixth power of mean flow velocity, and that the longitudinal vortex generated the aerodynamic sound larger than the transverse.

The sound induced by turbulence in an unbounded fluid is generally called aerodynamic sound. With respect to aerodynamic sound, Lighthill [

Regarding the vortex, emphasis has been placed especially on longitudinal vortex which rotates about the axis whose direction coincides with the flow direction. In the automobile industry the reduction of aerodynamic noise becomes more and more important for the comfortable vehicle since noises caused by engine, power train, tires, and other noise sources have been steadily reduced in recent years. It is well known that the front pillar of an automobile is regarded as one of the most dominant area in generating aerodynamic noise due to strong longitudinal vortices. Separated flows behind the front pillar generate the longitudinal vortices. Based on the theories as mentioned above, many researchers have so far tried to reveal the generation mechanism of aerodynamic sound. Haruna, Nouzawa, Kamimoto and Sato [

There have been so many studies to reveal the generation mechanism of aerodynamic noise produced by longitudinal vortex. However, it has not yet been clarified that how the longitudinal vortex system has been generated and how this system produces the aerodynamic noise. The final objective of our study, therefore, is to simplify this specific aerodynamic noise problem to obtain a thorough understanding of how the longitudinal vortex is produced, and how the noise can be estimated quantitatively.

Ogawa and Takeda [

rotating image of configuration of the vortex system. Chain line schematically shows rotating axes of the each vortex system. Two wings with the tip angles of 40 and 90 degrees show the longitudinal vortex system whose rotating axis is located in the flow direction has cone-shaped configurations. However, with the increase of tip angles, the cone-shaped configurations began to collapse around 110 degrees and over 120 degrees the vortex system shifted to the transverse vortex system whose shapes are no longer cone-shaped but elliptic with rotating axes vertical to the mean flow direction. This visualization method was able to clearly capture the change of vortex system from the longitudinal to the transverse vortex system. Although the spatial scale of the longitudinal vortex system is smaller than the transverse one, the rotating speeds of hydrogen bubbles in the longitudinal qualitatively seems to be faster than those in the transverse.

As a next step, CFD will be employed to analytically investigate the structure of the longitudinal vortex. The study uses the software STAR-CCM+ with software V10.04.009. In the simulation, numerical delta wing model has six tip angles of 40, 90, 110, 120, 130, and 140 degrees just as in flow visualization experiment.

This study employed RANS (Reynolds Averaged Navier-Stokes Simulation). This approach is valid when the maximum Mach number in the domain is less than 0.2 - 0.3. Eddy viscosity models use the concept of a turbulent viscosity to model the Reynolds stress tensor as a function of mean flow quantities. For the accurate simulation, Menter SST k-ω model was used as the hybrid models in the form that k-ω model was used close to the wall whereas the standard k-ε in the completely turbulent region was employed. The SST k-ω turbulence model is a two-equation eddy-viscosity model which has become very popular. The shear stress transport (SST) formulation combines the best of two worlds. The use of a k-ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k-ω model can be used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k-ε behavior in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties. It therefore follows that SST k-ω model often merit it for its good behavior in adverse pressure gradients and separating flow. In the steady state analysis, the maximum step number is 3500. Reynolds number is defined in Equation (1), where mean air flow velocity U = 10 m/s, representative length L = 0.26 m, and kinematic viscosity ν = 1.43 × 10^{−}^{5} m^{2}/s and results in Re = 1.7 × 10^{5}. This implies that the flow is turbulent and Mach number is 0.029.

flows can be maintained around the delta wing model. The attack of angles for all the models are 15 degrees the same as that used in the running water channel. Total number of mesh ranges from 7.48 million to 8.11 million, depending on the tip angles of the model. Minimum mesh size is 0.25 mm in closest vicinity to the leading edge of the model. In generating mesh, the prism layer meshing method was adopted. This meshing method was used to optimize the mesh size in the boundary layer as shown in

simulation with higher accuracy, the thickness of prism layer was decided so that wall

_{t} could be less than 1.0 in the section C. The configuration of the vortex still remains cone-shaped until 100 degrees. Around 110 degrees the vortex system becomes unstable. Over 120 degrees, the rotational radius of the vortex increases and the configuration of the vortex begins to shift from cone-shaped to elliptical. That is, the vortex generated behind the leading edge changed from the longitudinal vortex system to the transverse vortex system whose rotating axis is vertical to the mean flow direction. It follows, therefore, that the longitudinal vortex can remain until the tip angle of 100 degrees, and going through the transient region around 110 degrees, the longitudinal vortex has been changed to the transverse vortex over 120 degrees. _{p} is described in Equation (3) where P_{m }is the pressure on the wing surface and P_{∞} is the static pressure in the uniform flow speed U_{∞}.

C_{p} coefficients show how the pressure on the wing behaves, compared with the pressure in the uniform fluid flow. It, therefore, follows that if C_{p} has negative value, the pressure on the wing surface is lower than that in the uniform flows. The characteristics of the longitudinal vortex are well reflected by pressure coefficients on the wing surface. The fast flows close to the wing tip are rapidly separated, but the rotational radius is small. Since the centrifugal force has the property that it is proportional to the square of the velocity and is in inverse proportion to the rotational radius, this causes the tip of the longitudinal vortex to induce the largest negative pressure. As a result, the wing tip has the lowest pressure coefficients in dark green color for θ = 90 and 110 degrees as shown in

On the other hand, although the transverse vortex system is larger in spatial scale, the velocities of the particles are much slower than those of the particles in the longitudinal system. The numerical results agree well with the experimental ones obtained by flow visualization with the hydrogen bubble method in the running water channel. Due to the slow velocity of the fluid particles, the transverse vortex system causes smaller negative pressure on the wing surface, compared with the longitudinal system.

Although it is of great importance to investigate the relationships between the vorticity and the unsteady movement of the vortex systems in terms of aerodynamic sound generation, the characteristics of the longitudinal and the transverse vortex system were, as our first step, investigated to clarify the relationships between the typical vortex systems and the vorticity in detail. The vorticity of the longitudinal and the transverse vortex system were calculated for tip angles of the wing model with 40, 90, 110, 120, 130, and 140 degrees just as investigated in streamlines and C_{p} distributions. The vorticity will be described in Equation (4) to Equation (8) with respect to

the X, Y, Z coordinate axes and velocity vector

The measurements of aerodynamic sound radiated from the delta wings with tip angles of 40, 90, 110, 120, 130, and 140 degrees were conducted at the wind flow velocity of 10, 20, 30, and 40 m/s in the low-noise wind tunnel of Kyushu University.

The microphone used is NL-52 of RIONCo., which can measure the sound with the frequency ranging 20 to 20,000 Hz. The measured aerodynamic sound was sent to the personal computer by means of a data-logger; NR-500 and a measuring unit; NR-HV04 of KEYENCE Co. All the data of aerodynamic sound were measured in the form of Z characteristics without any filter. Sampling frequency is 20 kHz and number of data are 400,000. To avoid frequency aliasing, digital sampling of the signal must be performed at least twice as rapidly as the highest frequency expected. This critical sampling rate is known as the Nyquist frequency. Therefore Nyquist frequency is 10 kHz in this test.

Aerodynamic sound measured in the far field is defined as Sound Pressure Level; SPL (dB) in Equation (9), where p is RMS of sound pressure measured 1.5 m away from the delta wing by the microphone, and p_{0} is

reference sound pressure;

Fast Fourier Transform (FFT) analyses were conducted under the condition that Hann window was used as window function with 50 blocks for data numbers of 400,000, overlapping coefficients 50%, numbers of arithmetic mean 99, and frequency resolution 2.5 Hz.

velocity. This prominent frequency region is considered to be aerodynamic sound radiated from the vortex systems behind the leading edge of the delta wings. Focuses are, therefore, put on the sound pressure level of the prominent frequency region. Since it is well known that human ear is the most sensitive in frequency band between 2000 to 4000 Hz, attention is paid to the prominent frequency band f from 2900 to 4000 Hz for the airflow velocity of 40 m/s. Generally speaking, the frequency f of shedding vortices from a bluff body is proportional to flow velocity U and inversely proportional to representative length of the body L, using Strouhal number St. The relationship is described as follows.

In the study, frequency band f was chosen as standard values of f = 2900 - 4000 Hz at U = 40 m/s. Since the frequency of shedding vortices is proportional to the airflow velocity in accordance with Equation (10), frequency bands are set as 0.25f = 725 - 1000 Hz at 0.25U = 10 m/s, 0.5f = 1450 - 2000 Hz at 0.5U = 20 m/s, and 0.75f = 2175 - 3000 Hz at 075U = 30 m/s. These chosen frequency bands are depicted in red-colored zone in

Next step is to investigate the dependence of both aerodynamic sound produced by the longitudinal and the transverse vortex system on the airflow velocity. Aerodynamic sound is evaluated in the form of overall values SPL_{OA} given by summing up sound pressure level L_{i} of the frequency band in the red-colored zone for the respective airflow velocity, based on Equation (11).

The purpose of our research is to clarify not only the characteristics of aerodynamic sound but also generation mechanism of aerodynamic sound produced by the longitudinal and the transverse vortex system. As our first step, the characteristics of aerodynamic sound will be studied in the present paper. According to Equation (11), sound pressure levels were calculated for six delta wings and for mean airflow speeds as shown in

system increases in proportion to the 5th to the 6th power of mean airflow speeds, and that aerodynamic noise radiated from the longitudinal vortex system is greater than that from the transverse vortex system.

So far the two vortex systems have been investigated in terms of the configurations, vorticity, pressure coefficients on the wing surface, and aerodynamic noise experimentally and numerically. From now on, focuses are put on the generation mechanism of the aerodynamic noise from standpoints of the theoretical phase. As it is well known, Lighthill’s equation is described as Equation (12),

where T_{ij} is Lighthill stress tensor, ρv_{i}v_{j}: Reynolds stress.

The problem of calculating the turbulence generated sound is therefore equivalent to solving this equation for the radiation into a stationary, ideal fluid produced by a distribution of quadrupole sources whose strength per unit volume is the Lighthill stress tensor T_{ij}.

The formal solution of Lighthill’s equation is given as

In the study, the mean density and sound speed are considered to be uniform throughout the fluid and the variations in the density ρ within a low Mach number, high Reynolds number source flow are then of order ρ_{0}M^{2}.

Therefore,

region, it may also be shown that

Thus, if viscous dissipation is neglected we make the approximation, provided that

The far-field acoustic pressure

According to Howe [

It therefore follows that since aerodynamic sound in the far field can be expressed as a function of the vorticity instead of the Reynolds stress, the vorticity plays a crucial role in generating aerodynamic sound. Equation (19) involves a double time rate of change. This implies that the vortices which change rapidly in time give great contributions to generating aerodynamic sound.

Curle [_{d} from a surface element of diameter

which exceeds by an order of magnitude (_{d} radiated by the dipole is proportional to the sixth power of airflow velocity.

This equation indicates that total power radiated by the dipole sound source increase in proportion to the sixth power of the mean airflow velocity. The direct power Π_{q} radiated by quadrupole occupying a volume V_{0} in the absence of the body is given as in Equation (22). This is Lighthill’s eighth power law.

The sound produced by the turbulence near solid body S is therefore dominated by the dipole since M is nearly equivalent to 0.01 in this study, and as M approaches zero the acoustic power exceeds the quadrupole by a factor

Precisely how small M should be for this to be true depends on the details of the flow, which determine the approximate values of A and

How [

In the far field the acoustic pressure is given by the linearized approximation

where

The contribution to the sound from surface friction if normally of order

relative to the contribution from the volume vorticity, where

The next step of our study is, by means of the above vortex sound equation, to simplify this specific aerodynamic noise problem to obtain a thorough understanding of how the longitudinal and the transverse vortex system is produced, and how the noise can be estimated quantitatively.

Flows around the front pillar of an automobile are typical of a flow field with separated and reattached flow by a vortex system. It is known that the vortex system causes the greatest aerodynamic sound around a vehicle. The objective of the present study is to clarify the relationship between vortical structures and aerodynamic sound by the vortex system generated around the front pillar. The effects of the slant angles of the front pillars on the generation of the vortex systems were investigated by changing tip angles of the wings. The separation vortices behind the pillars generate the organized vortex systems, depending on the slant angles of the front pillars. The typical flows are reproduced by the three dimensional delta wings under the condition of low Mach numbers and high Reynolds numbers. The vortex systems were essentially reproduced by three-dimensional delta wings which consist of the longitudinal and the transverse system. The characteristics of the longitudinal vortex system were investigated in comparison with the transverse system. The results obtained are as follows.

1) The flow visualization experiment and the computational fluid dynamics (CFD) captured well the characteristics of the flow structure of the two vortex systems. These results showed that the longitudinal with the rotating axis along mean flow direction had cone-shaped configuration whereas the transverse with the rotating axis vertical to mean flow direction had elliptic one. Increasing the tip angles of the wings from 40 to 140 degrees, there first exists the longitudinal vortex system less than 110 degrees, with the transition region ranging from 110 to 120 degrees, and finally over 120 degrees the transverse vortex system appears.

2) The longitudinal vortex system with small tip angles of the wings less than 90 degrees has much faster streamlines due to the shape of the delta wing than those of the transverse one. The fast airflows cause the pressure on the wing surface to be much larger negative pressures, compared with those of the mean flow speeds. In the magnitude of the vorticity, the longitudinal vortex has much greater than the transverse vortex. The rotational velocity and vorticity have their largest values at the tip of the vortex and reduce downstream along the vortical axis. This resulted in inducing the largest negative pressure at the tip of the delta wing surface. Due to the slow velocity of the fluid particles, the transverse vortex system causes smaller negative pressure on the wing surface, compared with the longitudinal system.

3) The characteristics of aerodynamic sound radiated from the two vortex systems were investigated in low Mach number flows at high Reynolds numbers in the low-noise wind tunnel. As a result, it was found that the aerodynamic sound radiated from both the longitudinal and the transverse vortex system was proportional to the fifth from sixth power of mean flow velocity. The results almost agree with Curle’s dipole prediction.

4) The longitudinal vortex generates the aerodynamic sound larger than the transverse vortex system. This might be due to the fact that the vortex sound generation source div(ω×v) in the longitudinal vortex probably is larger than that in the transverse vortex system, as is the case with the comparison of steady vorticity for two vortex systems.

Shigeru Ogawa,Jumpei Takeda,Taiki Kawate,Keita Yano, (2016) Aerodynamic Sound Radiated from Longitudinal and Transverse Vortex Systems Generated around the Leading Edge of Delta Wings. Open Journal of Fluid Dynamics,06,101-118. doi: 10.4236/ojfd.2016.62009

Re: Reynolds Number

U: Flow Velocity (m/s)

L: Representative Length (m)

ν: Kinematic Viscosity (m^{2}/s)

ω: Vorticity (1/s)

θ: Tip Angle of the Delta Wing (degree)

Cp_{t}: Total Pressure Coefficient

C_{p}: Pressure Coefficient

P_{m}: Pressure of Measuring Point on the Wing (Pa)

P_{∞}: Pressure in the Uniform Flow (Pa)

ρ: Density of Fluid (kg/m^{3})

SPL: Sound Pressure Level (dB)

f: Frequency

T_{ij}: Lighthill Stress Tensor

ρv_{i}v_{j}: Reynolds Stress

M: Mach Number

Π_{d} : Total PowerRadiated by the Dipole

Π_{q}: Total Power Radiated by the Quadrupole

G(x,y, t-τ): Green’s Function

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