^{1}

^{*}

^{1}

The purpose of the paper is to clarify the mechanism of generation and collapse of a longitudinal vortex system induced around the leading edge of a delta wing. CFD captured well characteristics of flow structure of the vortex system. It is found that the vortex system has a cone-shaped configuration, and both 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. The collapse of the vortex system was also studied. The system can still remain until the tip angle of 110 degrees. However, between 110 degrees and 120 degrees, the system becomes unstable. Over 120 degrees, the characteristics of the vortex are considered to have converted from the longitudinal vortex to the transverse one.

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 on 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 areas 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. As a first step, the present paper aims to reproduce the longitudinal vortex generated behind the front pillar of a vehicle by a simple delta wing model, and to clarify the mechanism of generation and collapse of the longitudinal vortex system.

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 V9.04.009. In the simulation, tip angles of the wing model vary from 40 degrees to 140 degrees in every 10 degrees step.

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, and when unsteady wake effects are not important.

Eddy viscosity models use the concept of a turbulent viscosity to model the Reynolds stress tensor as a function of mean flow quantities. K-ω model was used as the eddy viscosity model. It therefore follows that the flows are three-dimensionally calculated in steady state with turbulent model.

In the steady state analysis, the maximum step number is 3500. The speed of the water flow is 0.4 m/s as in the running water channel experiment. Reynolds number is defined in Equation (1), where U = 0.4 m/s, L = 0.26 m, and ν = 1.28 × 10^{−6} m^{2}/s and results in Re = 8.1 × 10^{4}. This implies that the flow is turbulent and Mach number is 0.001.

To investigate the longitudinal vortex in more detail, the numerical simulation was conducted. As shown in

_{t} less than 0.8 in the section B. The streamlines of the vortex indicate the characteristics that the configuration is cone-shaped and the flow at the tip of the vortex is fastest, which is shown by red-colored streamlines. The flows are separating at the leading edge and rotating around the vortex axis upward. At the place of the tip flows firstly separate due to the wing configuration and flows are going upward with the flow rotating around the vortical axis. The fastest flow velocity decreases their velocity along the vortical axis. This is because there are strong interactions between flows and the wing body. As a result, fluid particles dissipate their energy due to viscosity and convection.

Vorticity in the longitudinal axis was investigated in each section of A, B, and C as shown in

This C_{p} coefficient shows how the pressure on the wing indicates, compared with the pressure in the uniform flows. It, therefore, follows that if C_{p} has negative value, the pressure on the wing surface is lower than one in the uniform flows.

Characteristics of the longitudinal vortex are well reflected by pressure coefficients on the wing surface. The fast flows at 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 as shown in

In order to clarify the collapse phenomena of the longitudinal vortex, tip angles of the wing model were varied from 40 degrees to 140 degrees at every 10 degrees. Vorticity was used as a physical quantity for evaluating the transformation of the vortex. The vorticity will be described in Equation (3) to Equation (7) with respect to X, Y, Z coordinate axes and velocity vector

In evaluating the transformation of the vortex, time-averaged vorticity was employed for the extracted region where total pressure coefficient Cp_{t} is less than 0.8 in each section A, B, and C. Cp_{t} was used as a physical quantity for identifying the region where the vortex exists.

The longitudinal vortex can still remain until the tip angle of 110 degrees. However, between 110 degrees and

120 degrees the longitudinal characteristics become unstable. Over 120 degrees the characteristics of the vortex are considered to have converted from the longitudinal to the transverse one. It follows, therefore, that the longitudinal characteristics of the vortex are converted to the transverse vortex.

In order to investigate qualitatively the configuration of the vortex generated behind the leading edge, the streamlines of the vortex were investigated for tip angles from 100 degrees to 140 degrees at every 10 degrees.

wing with the tip angle from 100 degrees to 140 degrees. The depicted region where a threshold value of Cp_{t} less than 0.8 in section B is established in order to extract the streamlines at the region which exists the vortex. The configuration of the vortex still remains cone-shaped until 100 degrees. At 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-shaped. That is, the vortex generated behind the leading edge changed from the longitudinal to the transverse vortex which rotates on the Y axis lying at right angles to flow direction. After the shift from the longitudinal vortex to the transverse vortex,

It follows, therefore, that the longitudinal vortex can remain until the tip angle of 100 degrees, and undergoing the transient region around 110 degrees, the longitudinal vortex has been changed to the transverse vortex over 120 degrees.

The mechanism of generation and collapse of a longitudinal vortex system induced around the leading edge of a delta wing was investigated. The results obtained are summarized as follows.

1) CFD captured well characteristics of flow structure of the longitudinal vortex system. It is found that the longitudinal vortex has a cone-shaped configuration, and both rotational velocity and vorticity have their largest values at the tip of the vortex and reduce downstream along the vertical axis. This resulted in inducing the largest negative pressure at the tip of the delta wing surface.

2) With respect to the vortical characteristics, the following was clarified. The longitudinal vortex can still remain until the tip angle of 110 degrees. However, between 110 degrees and 120 degrees, the longitudinal characteristics become unstable. Over 120 degrees, the characteristics of the vortex are considered to have converted from the longitudinal to the transverse one.

ShigeruOgawa,JumpeiTakeda, (2015) Mechanism of Generation and Collapse of a Longitudinal Vortex System Induced around the Leading Edge of a Delta Wing. Open Journal of Fluid Dynamics,05,265-274. doi: 10.4236/ojfd.2015.53028

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})