#### 2.1. Mathematical Model

The purpose of this work is to study the acoustic field around an Ahmed body using available CFD tools. For this reason solving the differential equation is required. Motion of fluid is described by continuity and Navier–Stokes Equations (

1)–(

4). In order to reduce computational time a few assumptions are introduced to the present model. Fluid is viscous, incompressible and Newtonian, and gravitation is negligible and in this work is neglected. Use of the Reynolds Averaged Navier–Stokes (RANS) turbulent model in the first stage significantly improves model convergence. The results from

$k-\omega $ Shear Stress Transport (SST) are used as an initial condition for transient simulation.

Navier–Stokes equation:

where:

${u}_{x},{u}_{y},{u}_{z}$—air velocity vector components

p—pressure

${\rho}_{p}$—air density

${\mu}_{p}$—dynamic viscosity

Received pressure and velocity distributions are used to calculate the acting drag force

${F}_{d}$. To compare results with other studies, the non-dimensional drag coefficient

${C}_{d}$ is calculated according to Equation (

5).

where:

${F}_{d}$—acting drag force

A—frontal area of the vehicle

u—relative air velocity.

The initial calculations are conducted using the

$k-\omega $ SST turbulence model. Transport equations for this model are presented in (

6) and (

7).

Transport equation for the turbulence kinetic energy k:

Transport equation for the specific dissipation rate

$\omega $:

where:

${\mu}_{t}$—turbulent viscosity

S—modulus of the mean rate-of-strain tensor

${F}_{1},{F}_{2}$—blending functions

$\kappa =0.41$—Karman constant

${\sigma}_{k,1}=1.176,\phantom{\rule{1.em}{0ex}}{\sigma}_{\omega ,1}=2.0,\phantom{\rule{1.em}{0ex}}{\sigma}_{k,2}=1.0,\phantom{\rule{1.em}{0ex}}{\sigma}_{\omega ,1}=1.168,\phantom{\rule{1.em}{0ex}}{\alpha}_{1}=0.31,\phantom{\rule{1.em}{0ex}}{\alpha}^{*}=1,\phantom{\rule{1.em}{0ex}}{\beta}_{\infty}^{*}=0.09,\phantom{\rule{1.em}{0ex}}{R}_{\beta}=8$

${\beta}_{i,1}=0.075,\phantom{\rule{1.em}{0ex}}{\beta}_{i,2}=0.0828$.

Further CFD calculations were performed using the LES turbulence model. The governing equations were obtained by filtered continuity and Navier–Stokes (

8)–(

11) equations. The Smagorinsky–Lilly model was used to model the eddy-viscosity

${\mu}_{s}$.

Filtered continuity equation for LES:

Filtered Navier–Stokes equation for LES:

where:

$\overline{{U}_{x}},\overline{{U}_{y}},\overline{{U}_{z}}$—filter velocity vector components

$\overline{P}$—filter pressure

${\mu}_{s}$—eddy-viscosity

$|\overline{S}|$—invariant of the filtered-field deformation tensor

${L}_{s}$—subgrid-scale characteristic mixing length

$\kappa $—Karman constant

d—distance to the closest wall

${C}_{s}$—Smagorinsky constant

$\mathsf{\Delta}$—local grid scale

V—volume of the computational cell.

In order to predict the acoustic field around an Ahmed body the extended form of Lighthill’s analogy (known as the Ffowcs Williams–Hawkings equation) is used according to Equation (

12). The searched parameters are the pressure fluctuations

${p}^{\prime}$, which are hidden behind the density fluctuation

${\rho}^{\prime}$.

Ffowcs Williams–Hawkings equation:

where:

${u}_{i}$—fluid velocity component in the ${x}_{i}$ direction

${u}_{n}$—fluid velocity component normal to the surface f

${v}_{i}$—surface velocity component in the ${x}_{i}$ direction

${v}_{n}$—surface velocity component normal to the surface f

${T}_{ij}$—Lighthill turbulence stress tensor

${P}_{ij}$—compressible stress tensor

$H\left(f\right)$—Heaviside function

$\delta \left(f\right)$—Dirac delta function

${\delta}_{ij}$—Kronecker delta

${c}_{0}$—speed of the sound

${\rho}^{\prime}$—density fluctuation.

Because the flow is incompressible it is possible to assume

${p}^{\prime}={c}_{0}^{2}\left(\rho -{\rho}_{0}\right)={c}_{0}{\rho}^{\prime}$. Furthermore, the velocity components normal to the surface are

${u}_{n}=0$ i

${v}_{n}=0$. Using the above assumptions, the Ffowcs Williams–Hawkings equations simplify to Equations (

13)–(

15).

As a result from the calculations, the pressure fluctuation

${p}^{\prime}$ is received as a function of time. The pressure fluctuation is used to calculate the sound pressure level

${L}_{sp}$ according to Equation (

16).

where:

${p}_{ref}=2\times {10}^{-5}\left[Pa\right]$—reference acoustic pressure

${p}^{\prime}$—pressure fluctuation.

#### 2.2. Numerical Model

In order to study the flow field around vehicles and phenomena of flow separations, in 1984 Ahmed et al. proposed a simplified geometry of a vehicle [

9]. The model is characterized by a rectangular shaped body with a rounded front and a slanted top part of the rear. The most common model also has four cylindrical pillars under the geometry (

Figure 1). To this day the model is in use as a reference geometry for scientist all over the world, and it is known as the Ahmed body.

The Ahmed body has many applications in the scientific community. Many scientific works are focused on the numerical investigation of aerodynamic parameters depending on the turbulence model used. Guilmineau et al. [

10] compared

$k-\omega $ SST, the Explicit Algebraic Stress Model (EARSM), Detached Eddy Simulation (DES) and Improved Delay Detached Eddy Simulation (IDDES) models. The last two are called hybrid models due to joining the advantages of the RANS model with the complexity of transient models. Investigation has been conducted for Ahmed body geometry with two different slant angles, 25 and 35 degrees. The best results were received from the IDDES approach. Serre et al. [

11] presented studies on a reference structure with a rear slant angle of 25 degrees. They performed four simulation processes using LES and DES models, achieving overestimates of the drag coefficient between 6–16%. Additionally, the mesh used in the simulation depends on the used turbulence model. Banga et al. [

12] used a

$k-\u03f5$ realizable model with a tetrahedral mesh. The studies concentrate on drag and lift coefficients depending on the rear slant angle with an inlet velocity of 40 m/s. Ten different angles were tested ranging from 0 to 40 degrees. The minimum drag coefficient of 0.235 was obtained for the slant angle of 7.5 degrees. Meile et al. [

13] presented studies on aerodynamic parameters for slant angles of 25 and 35 degrees. Numerical simulations used the Reynolds Stress Model (RSM) of turbulence for a velocity range of 10 to 40 m/s. The cut-cell mesh was implemented. Results were compared with experimental data. Thomas et al. [

14] compared five different turbulence models to compute airflow around an Ahmed body with a rear slant angle of 25 degrees. Turbulence models employed in these work were:

$k-\u03f5$ standard,

$k-\u03f5$ realizable,

$k-\omega $ SST, Spalart–Allmaras (SA) and Wray–Agarwal (WA). The WA model is a one-equation turbulence model implemented in the ANSYS Fluent software as a user-defined function [

14,

15]. The advantage of this turbulence model is less simulation time required per iteration.

In this paper, an Ahmed body is used in CFD and CAA analyses. In the first stage of this work the numerical model is validated. Experimental data from the literature are compared with the results received from CFD simulations. It is an important step in this work because the accuracy of the calculated acoustic field depends on the quality of the CFD solution (Dykas et al. [

16]). The next and main step is the calculation of the acoustic field around the reference structure and on this basis the prediction of the acoustic field around real road vehicles in the future.

#### 2.2.1. Geometry Model

The geometry model of Ahmed body consists of three parts: a fore body with rounded edges, a middle section with a rectangular cross section, and a slanted rear end with an angle of 25 degrees. The body is mounted on four cylindrical pillars. The Ahmed body was prepared as a 3D computer model using the part design tool in CATIA software. The dimensions of the reference structure are shown in

Figure 1.

#### 2.2.2. Discretization

The influences of computational domain discretization were investigated by Sosnowski [

17] and Mansour et al. [

18]. Comparing three types of grid elements, namely tetrahedral, hexahedral, and polyhedral meshes, a hexahedral mesh was found to be the least numerical diffusive. Furthermore, using this type of element helps with shorter convergence time. For this reason the computational domain was discretized using a hexahedron mesh that was manually implemented using ANSYS ICEM software (

Figure 2). The number of elements depending on air flow velocity ranged from 0.5 to 5.1 million. The boundary layer consisted of 15 elements in height, where the height of the first element as a nondimentional coefficient y+ for 97% of elements was below 1 (

17).

where:

${u}_{\tau}$—friction velocity

$\mathsf{\Delta}s$—wall spacing

${\tau}_{wall}$—wall shear stress

${C}_{f}$—coefficient

$Re$—Reynolds number.

The cell Courant number calculated from Equation (

18) for 99.9997% of elements was below 1. The minimum orthogonal quality was 0.323. Additionally, between the boundary layer and the domain a transitional layer was applied. The transitional layer consisted of 15 elements in height. The growth ratio between the elements in both layers was 1.15, while the aspect ratio did not exceed 60.

where:

${u}_{\infty}$—free stream velocity

$\mathsf{\Delta}t$—time step

$\mathsf{\Delta}x$—cell size.

#### 2.2.3. Boundary Conditions

A finite-difference approximation requires defining a solution at the domain boundaries to receive a unique solution. This case is an example of airflow around a vehicle model in a channel. In order to avoid interaction of the airflow structures caused by the Ahmed body shape with channel planes, it is necessary to ensure the appropriate size of the computational domain. For this purpose, a 1 L × 2 L × 8 L calculation area was created around the Ahmed body geometry, where L is the vehicle’s characteristic length of 1.044 m. The major turbulence and swirl structures of the airflow are observed in the wake behind the vehicle. This area is defined as five vehicle lengths in the x-coordinate. Because of a high pressure area at the front of the vehicle model, the distance between inlet plane and Ahmed body surface is defined as two characteristic lengths. Similar recommendations on the size of the computational domain are used by the European Research Community on Flow, Turbulence and Combustion (ERCOFTAC).

The computational domain with surfaces names is presented in

Figure 3. The no-slip condition was applied to the surface of the Ahmed body (Surface G) and bottom plane (Surface A). The front plane (Surface B), where the fluid enters the computational area, was selected as the inlet boundary condition. Depending on the simulation, air velocity was in range from 20 to 100 m/s. The pressure gradient was zero for this surface. The outlet boundary condition was employed on the plane where the air leaves the flow field (Surface E). A reference pressure of zero was assumed. Zero gradients were applied for other parameters. A symmetry condition was applied along the middle section of the body for plane

$y=0$ (Surface F). This results in reducing the number of elements and increasing the speed of calculations. The boundary condition with the coordinates of each surface are described in detail in Equations (

19)–(

44).

(Surface A) The bottom surface of the channel is defined as a stationary wall with a no-slip condition:

where:

${k}_{w}$—turbulent kinetic energy in the wall cell

${\omega}_{w}$—specific turbulence dissipation in the wall cell

${y}_{w}$—distance from wall to cell centroid.

(Surface B) At the inlet of the channel the air velocity is fixed to a constant value. Depending on the case the velocity is from 20 to 100 m/s:

(a) Studies of pressure and velocity distribution, streamlines in symmetry plane

$y=0$, comparison of vortex system (

Section 3.1), streamwise velocity profiles, drag and lift coefficient as a function of time (

Section 3.2), overall sound pressure level and A-weighting sound pressure level (

Section 3.4):

where:

$I=1\%$

$\frac{{\mu}_{t}}{{\mu}_{p}}$ = 10

(b) Investigation of drag coefficient as a function of the Reynolds number (

Section 3.3):

where:

$I=1\%$—turbulence intensity

$\frac{{\mu}_{t}}{{\mu}_{p}}$ = 10—turbulent viscosity ratio.

(Surface C) The side of the channel is defined as a symmetry:

(Surface D) The top of the channel is defined as a symmetry:

(Surface E) At the outlet of the channel the pressure is fixed:

where:

${u}_{avg}$—mean flow velocity

$I=5\%$—backflow turbulent intensity

$\frac{{\mu}_{t}}{{\mu}_{p}}=10$—backflow turbulent viscosity ratio.

(Surface F) The side of the channel with an Ahmed body contour is defined as a symmetry:

(Surface G) The surface of the Ahmed body is defined as a wall with a no-slip condition:

where:

$\overline{n}$—unit vector normal to surface.

#### 2.2.4. Initial Condition

To solve the parental equation for unsteady flow using the LES (Large Eddy Simulation) turbulence model, an initial condition was required. In the beginning for

$t=0$, the velocity vector at each point is equal to the velocity vector calculated in the steady-state using the

$k-\omega $ SST turbulence model (

45). Pressures at each surface element were calculated every two computational time steps. The maximum frequency related with the Nyquist frequency was 25000 Hz. These sound frequencies cover the full range of human hearing.

where:

i—grid index on the x axis

j—grid index on the y axis

k—grid index on the z axis.