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Analysis of Different Turbulence Models in Simulation of Hypersonic Flow in the Wind Tunnel

Mohammad Javad Ameri, Mohammad Reza Heidari, Hashem Nowruzi*, Amin Najafi

Maritime Engineering Department, Amirkabir University of Technology, Tehran, Iran

Abstract: In the current paper, we investigated the hypersonic flow in the wind tunnel at the Mach number 8. For this purpose, the effects and performance of three different turbulence models SST/k-ω, RNG/k-ε and Realizable/k-ε on flow simulation are evaluated. Mesh sensitivity analysis is conducted and our results are validated against the existing experimental data with good accordance. To better phenomenological study on the hypersonic flow behavior, distributions of static pressure, total pressure, Mach number and streamlines at nozzle, diffuser and test chamber are investigated. We found that the SST/k- ω turbulence model is more efficient and accurate compared to two other turbulence models in simulation of hypersonic flow in the wind tunnel at high Mach numbers.

Keywords: CFD, turbulence models, hypersonic flow, wind tunnel

Received September 12, 2019; Revised October 24, 2019; Accepted October 28, 2019

Citation: Mohammad Javad Ameri, Mohammad Reza Heidari, Hashem Nowruzi, Amin Najafi, Analysis of Different Turbulence Models in Simulation of Hypersonic Flow in the Wind Tunnel. American Journal of Mechanical Engineering, 7 (4). 172-180. January 2019. doi:10.12691/ajme-7-4-3

1. Introduction

Wind tunnels are one of the well-known technologies in field of aerodynamic experiments. Until now, different types of subsonic, supersonic and hypersonic wind tunnels are fabricated and tested. Along with an experimental tests in the wind tunnels, CFD simulations in the virtual (i.e., numerical) wind tunnels are an efficient tool for more investigation on aerodynamically problems.

Complex and fully turbulence flow regime is visible in the hypersonic wind tunnels (due to high Mach number) [1,2]. Therefore, the necessity of study on the different turbulence models in simulation of hypersonic flow in the wind tunnel is evident. The main reason for this investigation is to choose an appropriated turbulence model with proper accuracy and low computational costs in engineering designs. So, some turbulence model such as large eddy simulation (LES) and direct numerical simulation (DNS) aren’t investigated due to relatively high computational costs [3,4]. In addition, it should be noted that the use of more complex turbulence models is associated with greater difficulty in selecting fixed coefficients and may not be efficient for engineering design in commercial applications [5,6].

Until now, several studies are conducted to investigate the different types of turbulence models for flow simulation in the wind tunnels [7-16] and to investigate the turbulence models in high pressure medium [17,18,19]. In addition, Among the various formulations proposed for turbulent modeling, models based on the concept of eddy dissipation are more interested for scholars. According to the cited works, we found the lack of study on the effects of eddy dissipation turbulence models in simulation of hypersonic flows in the wind tunnels. Therefore, in the current study three different turbulence models of RNG\k-ε, Realizable\k-ε and SST\k-ω are investigated and compared in simulation of hypersonic flow (i.e., Mach number of 8) in the educational wind tunnel

2. Governing Equations

The governing conservation equations of mass, momentum and energy in form of Navier-Stokes are presented in Eqs. (1) to (3). State equation of considered ideal gas is also shown in Eq. (4).

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

where,is the stress tensor and in the eddy dissipation turbulence models, it has a following relation with turbulence viscosity:

[picture] | (5)

The main purpose of all eddy dissipation turbulence models is related to calculate the turbulence viscosity. In the two equations turbulence models, we have two transfer equations of turbulence kinetic energy and turbulence dissipation to calculate the turbulence viscosity. In the RNG\k-ε, turbulent kinetic energy (k) and turbulence dissipation rate (ε) are as follows [20]:

[picture] | (6)

[picture] | (7)

In the RNG\k-ε, the turbulence viscosity is:

[picture] | (8)

In the Realizable\k-ε, turbulent kinetic energy (k) and turbulence dissipation rate (ε) are as follows [21]:

[picture] | (9)

[picture] | (10)

In the Realizable\k-ε, the turbulence viscosity is:

[picture] | (11)

here, has following form

[picture] | (12)

In addition, may be defined as follows:

[picture] | (13)

In the Realizable\k-ε, Prandtl number is where, as sound velocity is:

[picture] | (14)

Here, is equal . In addition, we assume Prt=0.85. Thermal expansion coefficient is

[picture] | (15)

as thermal expansion separation may be defined as follows [22]:

[picture] | (16)

Here, is the turbulent Mach number with formulation of (sound velocity is ). In the SST\k-ω model, turbulent kinetic energy (k) and turbulence dissipation rate (ω) are as follows [5,6]:

[picture] | (17)

[picture] | (18)

In the SST\k-ω, the turbulence viscosity is:

[picture] | (19)

Please see Ref. [23] for more detail about these two equations turbulence models.

3. Physic of the Problem

3.1. Geometry and Boundary Conditions

Main computational domain of the considered wind tunnel are nozzle, test chamber and diffuser. The nozzle is responsible for generating the hypersonic flow up to Mach number 8. Test model and the measuring equipments are placed in the test chamber. The role of diffuser is the pressure recovery in order to properly discharge of the hypersonic flow.

Static and total pressures at the nozzle inlet are defined so that the Mach number is equal to the unit. The boundary condition at the output of the diffuser is the defined output pressure. In addition, other walls of the wind tunnels have no slip boundary condition.

Figure 1. Schematic and boundary conditions of considered hypersonic wind tunnel (all dimensions are in mm)

Table 1. Specifications of the wind tunnel

According to the axi-symmetry flow, the axis passed through the nozzle and the diffuser is selected as the axis of symmetry. Schematic of the wind tunnel and considered boundary conditions are shown in Figure 1. Specifications of the wind tunnel are presented in Table 1. Moreover, density based and steady second-order implicit solver is considered in Fluent [24].

In order to proper compare of our different simulations, we need to present the same conditions. To this accomplishment, similar shock system (i.e., separation point in the divergent section of the diffuser) is considered in all simulations

3.2. Mesh Sensitivity Analysis and Validation Study

We used structured mesh for our computational domain. Three different mesh resolutions including coarse mesh with 93925 cells, fine mesh with 235552 cells and finer mesh with 375700 cells are tested and compared. Numerical results of stagnation pressure and Mach number on the axis of symmetry, and stagnation pressure and Mach number at nozzle outlet for considered three different mesh resolutions are presented in Figure 2 and Figure 3, respectively. As may be seen in Figure 2 and Figure 3, numerical results of fine and finer mesh resolutions are approximately similar. Consequently, due to lower computational cost of the fine mesh compared to finer one, fine mesh with 235552 cells is selected.

Figure 4 shows schematic of selected mesh resolution. In addition, y+ distribution on the nozzle wall (in case of using SST/k-ω turbulence mode) is illustrated in Figure 5.

Now, to ensure the accuracy of our numerical procedure, we validated our results for model HB-2 (i.e., as a standard model suggested by Supersonic Tunnel Association (STA), See Figure 6) with AGARD results [25] and Saravanan et al. [26] data (conducted in hypersonic shock tunnel (HST2) in Indian Institute of Science).

Structured mesh in boundary layer around the HB-2 is shown in Figure 7. It’s notable that 122939 cells is used for computational domain around the HB-2.

Figure 2. Stagnation pressure and Mach number on the axis of symmetry for three different mesh resolutions

Figure 3. Stagnation pressure and Mach number at nozzle outlet for three different mesh resolutions

Figure 4. Considered structured mesh at (a) test chamber and (b) inlet of diffuser

Figure 5. Distribution of y+ on the nozzle’s wall (in case of using SST/k-ω turbulence model)

Figure 6. Test model of HB-2 [26]

Figure 7. Structured mesh around the HB-2

Figure 8 (a) and (b) shows the static pressure distribution around the HB-2 at Mach number 8 compared to measured results at HST2 [27]. As may be seen in Figure 8 (a) and (b), an appropriate accordance is achieved between our results compared to Ref. [27]. In addition, stagnation point with high pressure in the nose of the HB-2 model is detected. The shock wave and its wake with sudden pressure change is more visible in Figure 8 (c) and (d). This shock wave is also reported by Heidari et al. [28]. Figure 8 (c) and (d) shows the Mach number distribution around the HB-2 measured at HST2 [27] compared to our results at Mach number 8.

Calculated lift and drag coefficients of HB-2 (at angle of attack 0° up to 10°) under three different turbulence models of RNG\k-ε, Realizable\k-ε and SST\k-ω are compared to AGARD results [25] and Spalart-Allmaras results [27] in Figure 9 and 10, respectively. Higher accordance is achieved for SST/k-ω, while lower accordance is obtained for Realizable/k-ε. However, as may be seen in Figure 11, for pitching moment coefficient, RNG\k-ε and SST/k-ω turbulence models have higher accordance with experimental data [25,27].

Figure 8. Static pressure distribution around the HB-2 : (a) measured at HST2 [27] and (b) present study under the Mach number 8 and Mach number distribution around the HB-2 : (c) measured at HST2 [27] and (d) present study

Figure 9. Lift coefficients under three different turbulence models of RNG\k-ε, Realizable\k-ε and SST\k-ω compared to AGARD results [25] and Spalart-Allmaras data [27]

Figure 10. Drag coefficients under three different turbulence models of RNG\k-ε, Realizable\k-ε and SST\k-ω compared to AGARD results [25] and Spalart-Allmaras data [27]

Figure 11. Pitching moment coefficients under three different turbulence models of RNG\k-ε, Realizable\k-ε and SST\k-ω compared to AGARD results [25] and Spalart-Allmaras data [27]

4. Results and Discussion

In this section, performance of three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε in simulation of hypersonic flow in the considered wind tunnel are investigated. Figure 12 shows distribution of stagnation pressure on the axis of symmetry (through the wind tunnel from nozzle inlet to diffuser outlet). Value of stagnation pressure is approximately constant through the nozzle (X=0.2 m) for all tested turbulence models, while, SST/k-ω model compared to two other turbulence models provided higher stagnation pressure value in the test chamber. Value of stagnation pressure is dramatically decreased in the inlet of diffuser due to shock wave generation. As may be seen in Figure 12, approximately similar value of stagnation pressure is achieved for RNG/k-ε and SST/k-ω, while, numerical results of Realizable/k-ε have significant difference with two other models. So, one can be conclude that Realizable/k-ε isn’t an appropriate turbulence model for prediction of the stagnation pressure in hypersonic flow. In addition, the local position for generation of shock wave (in the inlet of diffuser) is correctly predicted by SST/k-ω turbulence model.

Figure 12. Distribution of stagnation pressure on the axis of symmetry (through the wind tunnel from nozzle’s inlet to diffuser’s outlet) for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 13. Distribution of static pressure at the inlet of the test chamber for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 13 shows distribution of static pressure at the inlet section of the test chamber for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε. As may be seen in Figure 13, the pressure profiles of RNG/k-ε and Realizable/k-ε are approximately similar. Due to high velocity gradient in the slip line across the nozzle outlet, we expect significant difference on static pressure at the upper wall and lower wall of the test chamber. This phenomenon is only predicted by SST/k-ω turbulence model. We presented the slip line (according to vorticity streamlines) through the test chamber for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε in Figure 14. According to Figure 14, physical reality of hypersonic flow through the test chamber is properly predicted by SST/k-ω model.

Figure 14. Slip line through the test chamber for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Distributions of static pressure and Mach number at the inlet of nozzle for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε are illustrated in Figure 15 and Figure16, respectively. As may be seen in Figure 15, the static pressure has higher value at the wall boundary layer compared to axi-symmetry line of the nozzle. Overall trend of Mach number is similar for all tested turbulence model, while, at the local position of X=0.03-0.04 (around the Ma=7.9), SST/k-ω turbulence model provided higher value compared to two other models.

Figure 15. Distribution of static pressure at the inlet of nozzle for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 16. Distributions of Mach number at the inlet of nozzle for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 17. Distribution of static pressure at the outlet of diffuser for different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 18. Distributions of Mach number at the outlet of diffuser for different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Distributions of static pressure and Mach number at the outlet of diffuser for three different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε are presented in Figure 17 and Figure 18, respectively. As may be seen in Figure 17, the static pressure has higher value at the axi-symmetry line of the diffuser compared to wall boundary layer (due to flow separation at the divergent section of the diffuser). According to Figure 17, Realizable/k-ε has inappropriate trend in the section of the diffuser outlet. In addition, pressure variations due to shock wake is more visible in SST/k-ω turbulence model. Figure 18 shows approximately similar value of Mach number for all tested turbulence models. Moreover, SST/k-ω turbulence model provided an appropriate prediction of flow separation (pressure gradients) on the wall of the diffuser (See Figure 19).

Figure 19. Distribution of static pressure on the diffuser for different turbulence models of SST/k-ω, RNG/k-ε and Realizable/k-ε

Figure 20. Mach number contour at the inlet of diffuser (in case of SST/k-ω model)

For better phenomenological study, Mach number contours at the inlet of diffuser (in case of SST/k-ω model) is shown in Figure 20. In addition, flow streamlines at the diffuser are shown for RNG/k-ε, Realizable/k-ε and SST/k-ω in Figure 21. As may be seen in Figure 20, generation of shock wave at the inlet of diffuser is observed. Based on Figure 21, due to hypersonic flow separation, we found some vortex generation on the upper wall of the diffuser. Local position of flow separation and pressure at the outlet of diffuser is compared between the three tested turbulence model in Table 2. According to Table 2, CFD results of RNG/k-ε and SST/k-ω turbulence are more similar together. It is notable that, other turbulence models should be tested in the future to select the best turbulence models for flow simulation in the hypersonic wind tunnel. Moreover, using aartificial neural networks [29,30,31,32,33] to predict of hypersonic flow in the wind tunnels can be used in our future studies.

Figure 21. Flow streamlines at the diffuser for (a) RNG/k-ε, (b) Realizable/k-ε and (c) SST/k-ω

Table 2. Local position of flow separation and pressure at the outlet of diffuser

5. Conclusion

We studied the effects of three different turbulence models of Realizable/k-ε, RNG/k-ε and SST/k-ω in simulation of hypersonic flow at high Mach number 8 in the wind tunnel. Mesh sensitivity analysis is done and good accordance is achieved between our results against the existing experimental data. Based on our CFD results, SST/k-ω turbulence model compared to two other models, provided more accurate hypersonic flow characteristics (i.e., distribution of static pressure, total pressure, Mach number and streamlines). Study on the performance of other turbulence models in simulation of hypersonic flow can be considered in the future investigation

Conflict of Interest

The authors have no competing interests.

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