Extension of the KDO Turbulence/Transition Model to Account for Roughness

Wall roughness significantly influences both laminar-turbulent transition process and fully developed turbulence. This work has developed a wall roughness extension for the KDO turbulence/transition model. The roughness effect is introduced via the modification of the k and νt boundary conditions, i.e., the wall is considered to be raised at an extra height. The equivalent roughness height is linked to the actual roughness height, and the ratio between them is determined by reasoning. With such a roughness extension, the predictions of the KDO RANS model agree well with the measurements of turbulent boundary layer with a sand grain surface, while the KDO transition model yields accurate cross-flow transition predictions of flow past a 6:1 spheroid.


Introduction
Computational Fluid Dynamics (CFD) is widely used as a predictive tool for fluid motions. Fluids in nature and engineering applications often interact with solid walls. The walls, either natural or man-made, are with roughness. Wall roughness significantly affects laminar-turbulence transition process. After transition, turbulence on a rough surface is enhanced compared with that on a smooth one, leading to higher skin frictions and heat transfer rates. Therefore, it is necessary to account for roughness in CFD simulations.
It is unrealistic to set up a graphical CAD model with every roughness-element for a CFD simulation. Thus, it is assumed that the roughness-element size in any direction is small compared with the boundary layer thickness so that, above the roughnesses, the flow is averaged over numerous roughness elements that exact location of which is not accounted for [1]. The wall presented in the computational domain is smooth, and the velocity on the wall is also zero. To account for the effect of roughness, the "equivalent sand grain" approach [2] is commonly employed. This approach links the real roughness height to an equivalent roughness height. According to Nikuradse's experiments [2], there are empirical correlations between the height of the equivalent sand grain and the real roughness shape. The correlations proposed by Dirling [3] and Grabow and White [4] established the general paradigm of wall roughness modeling. The basic idea is to mimic roughness effect by increasing the turbulent eddy viscosity, ν t , or turbulent energy, k, in the wall region. The increment is determined by the equivalent roughness height.
This equivalent sand grain approach is successful in predicting fully developed turbulence. However, to capture the laminar-turbulent transition process which is influenced by roughness effect, additional modeling technique must be supplemented. Xiang et al. [5] proposed a hypersonic cross-flow transition criterion considering surface roughness, yet no roughness modeling for fully developed turbulence was presented. Liu et al. [6] employed an additional transport equation of A r , the roughness amplification factor, to reflect roughness effect in transition process. Liu et al. [6] also employed the Wilcox wall boundary condition for ω [7] to introduce roughness effect in turbulent region.
According to the previous work, it is evident that two roughness corrections are needed: one on the turbulent equations (k, ω), and one on the transition equations (γ). The roughness correction on the turbulent equations is a relatively mature one which, in fact, leads to enhanced turbulent viscosity. The enhanced turbulent viscosity itself is capable of reflecting the premature transition process. Thus, an ideal turbulence models with wall roughness corrections should be capable of capturing the influence of roughness on the transition process, without additional roughness corrections to transition equations, and this work aims to develop such a model.
By extending Bradshaw's assumption down to the wall, the Reynolds stress constitutive relation is established as, τ 12 is the principal Reynolds shear stress, ij uu   is the Reynolds stress tensor, k is the turbulent kinetic energy, R b is Bradshaw's coefficient, and S ij is the mean strain rate tensor.
For the KDO turbulence model [8], the Bradshaw's coefficient reads, in which, r=μ t /μ is eddy viscosity ratio. The eddy viscosity ratio, a measurement of the intensity of turbulence, is a transport variable. Due to the transport properties of r, R b is capable of

Roughness extension
The ''equivalent sand grain approach" is employed here. The basic idea is that, according to experimental data [2], the log-law still holds in a turbulent boundary layer with wall roughness.
The difference lies in that, the y + -U + profile moves upwards, leading to a modified log-law, h s is the sand grain height. The position of the wall can be considered to be raised from y to y+d 0 . d 0 is the equivalent sand grain height, which is less than h s due to that there are spaces among sand elements. On the other hand, roughness enhances the turbulent viscosity to a value that is much larger than the molecular viscosity on the wall, and we have, Note that equation (14) assumes that the boundary layer is fully turbulent, so equation (14) might not be valid for transitional flows. The solution of equation (14) is, By substituting equation (13) As seen in figure 1, equation (20) is valid in the rage 0 <y + (1) <10 3 which covers the viscous layer, the buffer layer, and the log layer. Along with the increment of y + , the flow undergoes laminar state, laminar-turbulent transition, and turbulent state, indicating that equation (20) could potentially capture all the flow states. For a rough wall, k + | w in equation (19)  The KDO RANS model [8] with this roughness correction is termed as KDOR. The KDO transition model [9] with this roughness correction is termed as KDOR-tran.

Turbulent boundary layer
Blanchard [12] conducted experiments over various surfaces. The turbulence on a sand grain paper the average height of which is 0.425 mm is often studied as a benchmark test case. The case corresponds to a zero pressure gradient flow, with an external velocity of 45m/s. Since the experiment focused on the roughness effects on fully developed turbulence, the KDOR model is  Figure 1 shows the skin friction distributions on the wall. The results of KDOR agree well with the experimental data, with C r =0.35 which corresponds to sand grain surface. The classic results of the roughness-extended SA model [1], termed as Boeing, is also shown as a reference. The KDO model yields much lower skin friction which corresponds to a turbulent boundary layer on a smooth wall. The velocity profile at x=1.2m is shown in figure 2. The KDOR model yields a shifted log-law velocity profile which is a typical velocity profile with roughness.  [13] on flow past a 6:1 spheroid at various Reynolds number and attack angle. The long diameter was 2.4m and the short diameter was 0.6m. The roughness height, h s , was about 3.3μm. The attack angle is set to 15° and the Reynolds number based on the long diameter is set to 6.5×10 6 . The flow was nearly incompressible and the Mach number is set to 0.2. This test condition was extensively studied by various transition models. The inflow turbulence intensity, Tu, is a controversial one. In the literature, Tu can range from 0.1% to 1%. A discussion with the researchers from DLR confirmed us that Tu was about 0.2% which is employed here. Since the surface of the spheroid was polished rather than covered with sand grain, it is preferred that C r be 0.50. However, to explore the influence of C r on the transition pattern, values 0.35 and 0.00 are tested. C r =0.00 corresponds to a spheroid with a perfectly smooth surface. Since it is a cross-flow transition case, the KDOR-tran model is employed. The cross-flow transition pattern on the spheroid is illustrated by the skin friction contours. The experimental result is shown in figure 4. Figure 5, 6, and 7 show the results of KDOR-tran with C r equals 0.50, 0.35, and 0.00, respectively. It is clear that the cross-flow transition pattern agrees with the measurements very well when C r equals 0.50. However, the transition onset location is slightly delayed compared with the measurements. A slightly increased Tu could optimize the predictions, but this work insists the value of 0.2% provided by DLR. When C r reduces to 0.35, the cross-flow transition pattern begins to deviate from the true pattern. When C r reduces to 0.00, the transition onset locations are greatly delayed and the transition pattern differs a lot from the true pattern. It is necessary to point out that the Cr=0.00 and Tu=0.2% condition yields laminar flows on the surface, and the results in figure 7 corresponds to the Cr=0.00 and Tu=0.6% condition. According to the simulations, wall roughness does play an important role on the laminar -turbulence transition process, and it necessary to take into account the roughness effects in turbulence modeling. An extension of the KDO turbulence/transition model has been derived. It assumes non-zero viscosity and turbulent kinetic energy at the wall and change the definition of the wall distance, d. Thus, CFD code developers need only to alter the boundary conditions. The governing equations remains unchanged. Unlike the classic roughness extensions which utilize the altered log-law to calibrate empirical coefficients, this extension use reasoning as the empiricism. The ratios between effective roughness height and "sand grain rough height" are 0.35 and 0.50 for sand grain paper and polished surface, respectively. The ratios indicate that the two roughness heights are of the same order, so the ratios are reasonable. In addition, the ratios are both verified by the CFD simulations.
Test on a flat plate boundary layer with sand grain surface shows that the KDOR model can well predict full developed turbulence. Test on a spheroid with polished surface shows that the KDOR-tran model is capable of capturing roughness-influenced cross-flow transition. The two models both employ the new roughness extension. The key formula of the roughness extension employs a y + -k + distribution which is obtained from the DNS data of a smooth flat plate. Surprisingly, the formula works well for rough walls, which exhibits some universality. Therefore, this work has successfully developed a roughness extension for the KDO turbulence/transition model. With such an extension, the KDO model is capable of capturing not only fully developed turbulence, but also the influence of roughness on the transition process, without additional roughness corrections to transition equations.