The Effect of Turbulent Intensity on Friction Coefficient in Boundary-Layer Transitional Flat Plate Flow

Abstract

In this study, the effect of inlet-turbulence intensity on the friction coefficient for the transitional boundary layer has been investigated computationally. For this purpose, two equation turbulence models of Std. k-ε, RNG k-ε, Std. k-ω, and SST k-ω have been compared with the Gamma–Theta (GT) transitional model, and it has been found that the Gamma–Theta model is the most consistent model with the experimental values of the ERCOFTAC T3A test case. Then, the effect of inlet-turbulence intensity on the friction coefficient has been computed by using this Gamma–Theta model. The transition from laminar to turbulence is shortened with increasing turbulence intensity by changing it from 1% to 10%. The most suitable inlet-turbulence intensity value with the experimental results of the ERCOFTAC T3A test case is found as Tu = 3.3%.

1. Introduction

Flow over a flat plate has many applications and is one of the basic fluid dynamics and heat-transfer topics for a better understanding of flows. It also has an important place in boundary-layer flows because it is a zero-pressure gradient flow, allowing for simplifications. Long and flat surfaces are encountered in many real flows. Examples of situations where the boundary layer forms over long and flat surfaces are flow over a ship’s hull, a submarine hull, aircraft wings, textile surfaces, and atmospheric flow over flat terrain. The flow characteristics over long and flat surfaces in engineering problems are similar to the flow over a flat plate. Therefore, understanding the flow over a flat plate is important in terms of understanding the flow characteristics over flat and long surfaces in engineering problems.

In addition to earlier studies [1], relatively earlier studies on flat plates have shown that inlet-turbulence intensity has a significant effect on the transition from laminar to turbulent, but that it increases heat transfer only slightly in the laminar region and that the turbulent boundary layer is only slightly affected [2,3]. More recent studies have shown that high-inlet-turbulence values produce significant increases in heat transfer [4,5,6]. However, it has also been reported that inlet-turbulence intensity has no effect in the laminar region [7,8].

The characteristics of laminar and turbulent boundary layers are different. Transition from laminar to turbulent, or relaminarization, can cause significant changes in the aerodynamic performance of the object. The aerodynamic forces and moments or heat transfer acting on the object in the fluid are directly affected by the position of the laminar to turbulent transition. In the laminar to turbulent transition region, a sudden increase in drag force and heat transfer occurs.

In order to reduce the drag force in the gas turbine airfoil, it is necessary to move the transition from laminar to turbulent to a certain distance from the leading edge. Sometimes, at low Reynolds numbers, the laminar region is extended in low-inlet-turbulent flow, and flow separation occurs in the laminar region before it becomes turbulent, which is not preferred in terms of fuel economy. As a result, even a 1% saving in fuel is very important economically [9]. The transition from laminar to turbulent can occur in 80% of the blade in low-pressure turbine blades if the inlet-turbulence intensity is low [10]. Therefore, it is important to be able to correctly solve the transition from laminar to turbulent.

There are many factors that affect the transition to turbulence. These are pressure gradient, Reynolds number, surface roughness, heat transfer, and inlet-turbulence intensity. These effects cannot fully represent all types of situations with the theoretical or empirical expressions presented previously [11]. Therefore, it is a subject open to study.

For successful numerical computations, an accurate prediction of the laminar to turbulent transition in a flat plate is necessary. We can divide the laminar to turbulent transition mechanism into three: natural transition, bypass transition, and separated-flow transition. We encounter the first two of these transitions in the flow over a flat plate (zero-pressure gradient flow). The characteristics of all three transition mechanisms are explained by Mayle [12]. Natural transition occurs with the formation of Tollmien–Schlichting waves in the laminar boundary layer. It can be estimated by linear stability analysis [13]. Bypass transition occurs with the formation of direct turbulence spots in the laminar boundary layer and at high-inlet-turbulence intensities (Tu > 1%). It is also explained analytically by the turbulence spot theory [1].

Generally, in transition, predictions are encountered mainly with two concepts. The first is the use of the turbulence models, and the second is the use of experimental correlations [14].

Radmehr and Patankar [15] have investigated the three well-known low Reynolds number (LRN) turbulence models, Launder and Sharma (LS), Lam–Bremhorst (LB), and Biswas and Fukuyama (BF) models, and they have been concluded that none of them is able to predict the quantitative aspects of transition correctly by comparing them with the ERCOFTAC T3A test case. A new low Reynolds turbulence model has been proposed by them for the prediction of transition on gas turbine blades, and they have obtained important improvements [16].

Lardeau et al. [17] have proposed a low Reynolds number nonlinear eddy viscosity model, and they have expressed that a substantial improvement in the representation of the transition process has been achieved.

Menter et al. [14] have introduced a correlation-based transition model using local variables and have applied this new model to various test cases (Langtry [18]). They have mentioned that the basic model framework (transport equations without the correlations) will be called the Gamma–Theta (γ-Re_ɵ) model.

The V2F turbulence model has been tested by comparing the results with the ERCOFTAC T3B case by Luo and Razinsky [19]. They have concluded that the V2F model predicts an earlier onset of transition compared to the data.

Another new transition model that uses the RANS (Reynolds Averaged Navier–Stokes)-based intermittency transport equation, coupled with the Wilcox low Reynolds k-ω model with no modification, has been proposed by Akhter et al. [20]. Some improvements have been obtained in the prediction accuracy for high-freestream turbulence cases that are commonly found in gas turbines.

Langtry [18], in his doctoral study, modified the SST turbulence model for unstructured mesh structures and presented a correlation-based turbulence model using local variables that can solve the turbulence transition region more accurately. This model was soon included in the ANSYS CFX R13 commercial CFD code. Langtry conducted validation studies on flat plates, helicopter aerodynamics, aircraft wing aerodynamics, and turbine blade aerodynamics in his doctoral study and demonstrated that this model was more successful than Menter’s SST model in the transition region. After this turbulence transition model was integrated into the ANSYS CFX code under the name Gamma–Theta, it was integrated into the other most well-known CFD codes in the market, ANSYS Fluent and Star CD, and became the most widely used turbulence model in the industry for solving the transition region. In the literature, the Gamma–Theta model has been used and validated in studies numerically examining the transition region [21,22,23].

Bochon et al. [21] stated that the turbulence model that best solves the conjugate heat transfer in the transition region from laminar flow to turbulence is Gamma–Theta and that it is possible to obtain successful results by modifying other two-equation models like Gamma–Theta.

Piotrowsky et al. [22] studied the Gamma–Theta transition turbulence model on a plane plate and N3-60_04 turbine airfoil. As a result of the study, they stated that the Gamma–Theta turbulence model is in good agreement with the experimental results and can solve the transition to bypass wake-induced turbulence very well.

Suluksna and Juntasaro [23] studied to determine the effects of inlet-turbulence intensity and pressure gradient on the transition region flow characteristics numerically using the Gamma–Theta model. They stated that the critical Re number changes only with the turbulence intensity, and that the pressure gradient does not affect the critical Re number much in zero pressure gradient and favorable pressure gradient flows, and the pressure gradient effect can be neglected.

Another method used in transition predictions is the use of empirical correlations:

Abu-Ghannam and Shaw [24] proposed a correlation between the Reynolds number of the transition to turbulence and the inlet-turbulence intensity, depending on the boundary-layer momentum thickness. The correlation is expressed as follows:

$$R{e}_{\theta t}=163+\exp \left(6.91-Tu\right)$$

(1)

The correlation suggested by Mayle (1991) [12] is as follows:

$$R{e}_{\theta t}=400 T{u}^{-0.625}$$

(2)

Suzen and Huang [10] included the following expression for the transition to turbulence in their numerical model:

$$R{e}_{\theta t}=120+150 T{u}^{-2/3}$$

(3)

The correlations expressing the turbulent transition Reynolds number according to the inlet-turbulence intensity proposed by Abu-Ghannam and Shaw [24], Mayle [12], and Suzen and Huang [10], as mentioned above, are compared in Figure 1.

Taghavi–Zenous et al. [25] experimentally investigated the laminar-to-turbulent flow on a flat plate at different inlet-turbulence intensities. The measurements were taken with a hot wire velocimetry in the range of 1.5–4.4% inlet-turbulence intensity. They stated that with the increase in inlet-turbulence intensity, the transition region becomes smaller and regresses towards lower Reynolds numbers, and they derived an empirical expression for the turbulent transition Reynolds number depending on the inlet-turbulence intensity as follows:

$$R{e}_{\theta t}=\sum a_i . e^{-\left(Tu-b_i/c_i\right)} i=1, 2, 3$$

(4)

where: $a_i$: 3346, 879.9, 419.5; $b_i$: −0.3656, −0.8927, −12.93; $c_i$: 0.2825, 1.782, 18.29.

Various studies on boundary-layer transition are ongoing. A solver for boundary-layer transition was developed by Bhushan and Walters [26]. Fabrizio [27] investigated the critical phenomena in the laminar–turbulent transition with a mean field model. A new turbulence model for modeling turbulent transition according to mechanical priorities was proposed by Vizinho et al. [28]. The effects of wall heating on the boundary-layer transition on a flat plate were investigated numerically by Subaşı and Güneş [29]. An evaluation of Reynolds Averaged Navier-Stokes (RANS Reynolds Averaged Navier-Stokes) turbulence models in convective heat transfer was made numerically by Abdollahzadeh et al. [30]. A new laminar kinetic energy model for the prediction of pre-transition velocity fluctuations and the boundary-layer transition was presented by Medina et al. [31]. Lienhard [32] proposed a correlation for heat transfer in plane plate boundary layers under laminar, transitional, and turbulent flow conditions. Dotto et al. [33] performed large-eddy simulations (LES) for a zero-pressure gradient boundary layer developing over a flat plate to investigate the transition mechanism for variable free-stream turbulent properties. A comparative study of experiments with direct numerical simulations (DNS) of free-stream turbulence transition has been performed by Mamidala et al. [34]. Transition under free-stream turbulence, together with large integral-length scales, has been investigated by Duravic et al. [35] in another DNS study.

As can be understood from the above studies, the findings in some studies contradict each other, and since the effect of external turbulence on the flat plate is encountered in important application areas, studies on this subject are needed. Since the flow on the flat plate is frequently encountered in engineering problems, it is important to develop basic flow equations, such as the boundary layer for flow on the flat plate, to control the boundary layer and examine the transition to turbulence. In this study, the effects of the inlet-turbulence level on the zero-pressure gradient flow on the flat plate based on the ERCOFTAC T3A test setup were investigated by comparing various turbulence models, and it was determined that the Gamma–Theta (GT) model was the most suitable model. Then, using this model, the effect of turbulence intensity on the friction coefficient (Cf) was calculated numerically by keeping the integral length scale constant, and the results were provided in comparison with the experimental results.

2. Materials and Methods

In this study, the computational fluid dynamics (CFD) method is used to investigate the effect of inlet-turbulence intensity on the friction coefficient for the zero-pressure gradient (ZPG) flow, and for this purpose, conservation equations, namely continuity, momentum, and turbulence model equations, are solved under specified boundary conditions using ANSYS CFX commercial code. If the standard deviation is divided by the mean velocity ($\overline{u}$), the turbulence intensity, Tu, is obtained:

$$Tu={\displaystyle \frac{u^{\prime }}{\overline{u}}}\times 100 \%$$

(5)

where the velocity fluctuations, $u^{\prime }$, are in the main direction of the flow. The turbulence-length scale is considered as the sum of the integral length scale and the micro scale. Std. k-ε, RNG k-ε, std. k-ω, SST k-ω, and Gamma–Theta (GT) have been used and compared. The Gamma–Theta turbulence model is a turbulence model that solves two separate transport equations for the transition criterion and “intermittency” based on the Reynolds number defined according to the momentum thickness, thus solving the transition region much more precisely. The empirical correlation it uses is designed to capture the bypass transition even at low-inlet-turbulence intensities [36]. It is used with the SST turbulence model in ANSYS CFX. Since all turbulence models are solved using their standard forms and recommended turbulence coefficients in the program, detailed information about the turbulence models and coefficients used can be found in the work of Erşan [37] and in the modeling manual of the program (ANSYS CFX R13) [38]. ANSYS CFX includes a powerful pre- and post-processing interface and an advanced finite volume solver. ANSYS CFX solver has proven its reliability in terms of the results it has provided, both in the industry and in the literature. Stable results can be obtained during the solution thanks to its coupled solver.

2.1. Geometrical Model and Boundary Conditions

In the flat-plate calculations, the computational domain of H = 0.26 m and L = 2 m, shown in Figure 2 below, was used. This geometry was modeled under the same conditions, taking the ERCOFTAC T3A experimental setup as a reference (ERCOFTAC). A no-slip condition was defined at the lower plate to represent viscous flow over a flat plate. A frictionless wall-boundary condition was defined at the upper plate. At the outlet, the pressure was defined as zero. Uniform inlet velocity of 5.3 m/s was defined at the inlet. These boundary conditions were also used to represent the ERCOFTAC T3A test case. In all compared turbulence models, GT, std. k-ε, RNG k-ε, std. k-ω, inlet-turbulence intensity (Tu), and length scale (LS) have been taken as 3.3% and 0.026 m. Inlet-turbulence intensities have been taken as 1%, 2%, 3.3%, 5%, and 10% for fixed-length scale of 0.026 m in the cases where the effect of turbulence intensity was examined. Wall-friction fluctuations, non-uniformities in uniform inlet velocity, and small residual gradients in the zero-pressure-gradient (ZPG) flow conditions in the represented test case have been neglected.

2.2. Mesh Independence Study

In the computations, the mesh-independent solution was investigated for four different network structures. The mesh-independent solution was obtained in the fourth mesh structure, and the computations were repeated with this mesh structure. The mesh-independent solution was obtained in the mesh structure with 60,000 nodes. Four different mesh results are shown in Figure 3.

The computational domain seen above is divided into meshes with high-quality hexahedral solution elements, as seen below in Figure 4a. A total of 135,610 nodes were used. Dense elements were used at the entrance, near the wall, and leading edge of the plate, as also seen from Figure 4b,c. Maximum y⁺ values are less than 3.

2.3. Thermophysical Properties

The thermophysical properties of air have been considered at the inlet temperature and are provided in

Table 1. Thermophysical properties of air.

Density	1.116 kg/m3
Viscosity	1.725 Pa.s
Conductivity	0.02428 W/mK
Specific heat	1003.8 J/kgK

below [39].

3. Results and Discussion

Laminar and turbulent numerical solutions for the ERCOFTAC T3A experimental test case on a flat plate are provided and discussed in Figure 5 below.

When defining the Reynolds number in the flow on a flat plate, the characteristic length is used as the length in the flow direction. In Figure 5, from the experimental results, we see that the transition to turbulence begins at a local Re number of approximately 140,000 in the flow, and that the flow becomes fully turbulent at a local Re number of approximately 270,000.

In Figure 5, it is seen that the laminar numerical solution provides results that are inconsistent with the experimental data, as expected, in the turbulent transition and turbulent flow regions. It is seen that the results obtained using the Std. k-ɛ, Std. k-ω, SST k- ω, and RNG k-ɛ turbulence models are also inconsistent in the laminar and turbulent transition regions, as expected. As shown in Figure 5, the Gamma–Theta-SST turbulent transition model also agrees with the experimental results in the turbulent transition region, besides good agreement in both the laminar and the turbulent regions.

The above results show that the Gamma–Theta model is a suitable model for solving laminar, turbulent transition, and turbulent regions in flat-plate flow. However, although the compatibility with the experimental results in the turbulent transition region is not as good as in the laminar and turbulent regions, it can be said that it is at a quite acceptable level when the complexity of the flow in the transition region and the fact that other turbulence models cannot capture this region at all are taken into account. In the transition region, the Gamma–Theta model starts the transition earlier than the experimental data and completes the transition a little earlier. The friction coefficient is estimated to be, at most, 40% higher than the experimental results in this region. These results are for the ERCOFTAC T3A inlet-turbulence intensity Tu = 3.3% experimental case, and the turbulence-length scale is not specified in this experimental case; the length scale is taken as LS = 0.026 m in the calculations in this study. A certain part of the approximately 40% deviation from the experimental results in this region can be attributed to this.

When we inspect Figure 5, if the flow was completely turbulent (Re_x > 270,000), it is seen that the Gamma–Theta model is the most suitable turbulence model, and the model with the same suitability as this model is the Std. k-ω model. The other most suitable models are the SST- k-ω, Std. k-ɛ, and RNG k-ɛ models, respectively. However, although the Gamma–Theta and Std. k-ω models are almost in agreement with the experimental data, the other models predict the friction coefficient below the experimental data at Reynolds numbers smaller than Re_x = 440,000, and it can be said that all of them show almost the same performance. Again, if the flow was completely laminar, it is seen from Figure 5 that the numerical solution is in very good agreement with the experimental data in the laminar region (0 < Re < 170,000).

The Gamma–Theta turbulence model, which is the most suitable model to investigate the effect of inlet-turbulence intensity, was selected, and the effect of inlet-turbulence intensity was investigated with this model. Five different calculations were made by keeping the turbulence-length scale constant and changing the inlet-turbulence intensity between 1% and 10%. The results are shown in Figure 6a, together with the experimental and laminar numerical solution results. The turbulent transition regions are shown in

Table 2. Transition regions of inlet-turbulence intensity to turbulence on the plane plate in the Gamma–Theta turbulence model (this study).

Tu (%)	LS (m)	Inlet Velocity (m/s)	Laminar Region	Transition Region of Turbulence	Turbulent Region
1	0.026	5.3	0 < Re < ∞	-	-
2	0.026	5.3	Re < 280,000	280,000 < Re < 400,000	Re > 400,000
3.3	0.026	5.3	Re < 115,000	115,000 < Re < 260,000	Re > 260,000
5	0.026	5.3	Re < 60,000	60,000 < Re < 150,000	Re > 150,000
10	0.026	5.3	-	-	0 < Re < ∞

. In addition, the results of the experimental study of Taghavi–Zenous et al. [25] are provided in Figure 6b for comparison. The experimental conditions of this experimental study are also shown in

Table 3. Experimental conditions [25].

Deney Kodu	Tu (%)	LS (m)	Inlet Velocity (m/s)
ZPG-TA	1.5	0.009	15.5
ZPG-TB1	3.2	0.0105	12
ZPG-TB2	3.3	0.011	15
ZPG-TC1	4.3	0.0126	8.35
ZPG-TC2	4.4	0.0129	10.5

.

Since the transition to turbulence in the flow occurs when the critical Re number is exceeded, the change in the turbulence transition region according to the Re number is provided in Table 2 for different inlet-turbulence intensities. As the inlet-turbulence intensity increases, the transition to turbulence occurs at lower Re numbers. In the case where the inlet-turbulence intensity is 1%, since the calculations are made in the range of 0 < Re < 700,000 and the turbulence transition region is outside this range, the turbulence transition region could not be determined and is shown with a “-” in Table 2. In the case where the inlet-turbulence intensity is 10%, the flow occurs directly as turbulent, and this time, there is no laminar and turbulent transition region, so it is shown with a “-” in Table 2.

As seen in Figure 6a, with the increase in the inlet-turbulence intensity, the transition to turbulence became easier, and the laminar region shortened. Some results in Figure 6a are summarized in Table 2, and it can be seen that as the turbulence intensity increases, the transition region to turbulence narrows. This situation was also determined in the experimental study of Taghavi–Zenous et al. [25].

When Figure 6a,b are examined, it can be easily said that the numerical results in this study are consistent with the experimental results. Although the turbulence length scale was kept constant in this study, it should not be forgotten that this consistency should be evaluated qualitatively since it was not kept constant in the experimental study of Taghavi–Zenous et al. [25]. Because, as stated in the studies of Hancock and Bradshaw [40] and Iyer and Yavuzkurt [41] where their studies were evaluated, the turbulence-length scale is also effective on the boundary layer, along with the turbulence intensity. Jonas et al. [42] stated a significant effect of turbulence-length scale on the onset and the end of the bypass transition. But, in their study that investigates boundary layers corresponding to the ERCOFTAC test case TA+ for fixed Tu = 3% and an LS range of 2.2 mm–33.3 mm, Cf distribution does not affect length-scale (LS) changes above the LS = 15.3 mm. By representing ERCOFTAC test cases T3A, T3B, and T3AM for transitional integral parameters, Darag and Horak [43] found that the transition location for the Tu level higher than 4% is only a function of turbulence intensity (Tu) and unaffected by significant changes in the length scale.

Also, the importance of wall-friction fluctuations has been emphasized by Jonas et al. [42], which has been considered by the ERCOFTAC test case T3A+. But there is no information about experimental uncertainty for the friction coefficient, and the possible effects of this type of fluctuation must be kept in mind.

4. Conclusions

Two-equation turbulence models, Std. k-ε, RNG k-ε, Std. k-ω, and SST k-ω, have been compared with the Gamma–Theta (GT) transitional model, and it has been found that the Gamma–Theta model is the most consistent model with the experimental values of the ERCOFTAC T3A test case.

The transition from laminar to turbulence is shortened with increasing turbulence intensity by changing it from 1% to 10%.

In the case where the inlet-turbulence intensity is 1%, the flow occurs directly as laminar, and this time, there is no laminar and turbulent transition region.

It has also been observed that with the increase in the inlet-turbulence intensity in the flow over the plane plate, the transition to turbulence becomes easier, the laminar region shortens, and the transition region to turbulence narrows as the turbulence intensity increases. These results are also supported experimentally since they were also found in the experimental study of Taghavi–Zenous et al. [25].

In the case where the inlet-turbulence intensity is 10%, the flow occurs directly as turbulent, and this time, there is no laminar and turbulent transition region.

The most suitable inlet-turbulence-intensity value with the experimental results of the ERCOFTAC T3A test case is found as Tu = 3.3%.