The influence of boundary condition functions on the quality of the solution and its sensitivity to coefficients of k-ε turbulence models

The paper is devoted to the problem of boundary conditions influence on the quality of the solution obtained with use of k-ε turbulence models. There are calculation results for different boundary conditions and two methods: standard k-ε and RNG k-ε in the paper. The flow parameters obtained from the calculation are compared with our own measurement results. Moreover, the influence of input data on the inflow edge on sensitivity coefficients is shown and analysed in the paper. The research is performed for components of velocity and turbulence kinetic energy.


Introduction
The problem of the solution quality is described in many papers, for example in Hrenya et al. [4], Shih et al. [5] and Shimada and Ishihara [6] which are concerned with the analysis of model coefficients or improving the followed turbulence model based on the k-ε method.This paper is devoted to the problem of boundary conditions influence on the solution quality and its sensitivity to coefficients of the k-ε method.The aim of the paper is to show the importance of the correct description of boundary conditions at the inflow edge of a calculation domain.
The inflow parameters are usually described at inflow edges by a few functions of flow parameters.The solution results depend on both the values of inflow parameters and their derivatives.Lack of fulfillment of derivatives continuity significantly influences the obtained solution.Here, the problem is presented for a twodimensional incompressible steady flow around a square cylinder.The research is limited to the following flow parameters: components of velocity and turbulence kinetic energy.

Description of the research problem 2.1. Research methods
The subject of this research is a two-dimensional incompressible steady flow around a square cylinder.The set of this model in a calculation domain is shown in Fig. 1.The mesh of FVM contains 80075 cells and it is more dense on walls and around the square.Calculations have been made using the Fluent program for two versions of the k-ε models: standard and RNG ones.The assumed model coefficients are as follows: C ε1 = 1.44,C ε2 = 1.92,C µ = 0.09, σ k = 1.0 and σ ε = 1.3 for the standard k-ε model and C ε1 = 1.42,C ε2 = 1.68 and C µ = 0.0845 for the RNG k-ε model.
In order to check the quality of calculation results measurement results from the wind tunnel are used.These measurements have been carried out in the wind tunnel of the Wind Engineering Laboratory in Cracow University of Technology by the author with associates.The model was set at the ground of the wind tunnel.The length of the model was b = 2050 mm and the flow in the middle-plane may be treated as two-dimensional one.The research in the wind tunnel is described in the following papers: Błazik-Borowa [1] and Błazik-Borowa et al. [2].
The equations of the k-ε methods contain semi-empirical coefficients which significantly influence the calculations results.The sensitivity analysis serves to check the influence of small changes of the model parameters on the problem solution and it is described using sensitivity coefficients, which may be calculated from formula: The methods of calculations of sensitivity coefficients, examples of sensitivity analysis and applications of the sensitivity analysis results are presented in the papers by Błazik-Borowa [1] and Błazik-Borowa et al. [3].Sensitivity coefficients depend on approximation methods, the quality of the mesh, etc.Here, the consequence of the boundary conditions is checked for the sensitivity of flow parameters on model coefficients.

The description of inflow boundary conditions
The boundary conditions are located as it is shown in Figure 1.The input flow parameters may be given only at the edge with the boundary condition called inlet velocity.The calculations have been made for four sets of functions of the inflow parameters: Case No 1 Parameters are constants and they are equal to values measured in the middle part of inflow plate of the wind tunnel, i.e: input velocity u o = 10 m/s, turbulence intensity I u = 0.052, turbulence kinetic energy k = 0.2704 m 2 /s 2 and dissipation of turbulence kinetic energy ε = 0.0162.

Case No 2
The functions of velocity u o and turbulence intensity I u are determined on the basis of measurement results.Velocity is described by the following relationships: (2) where u h = 9.67 m/s -velocity at the upper edges of the square, h = 20 cm -height of the square, H = 142 cm -height of the working section of the wind tunnel, δ = 25 cm -thickness of the boundary layer.
The turbulence intensity is expressed by the functions: 0 87 1 0 63 1 0 15 0 24 1 0 18 1 0 07 0 6625 0 6625 . .m and m (5) Turbulence kinetic energy and its dissipation are calculated on the basis of velocity and turbulence intensity on the basis of the following formulae: where L x = 142 cm -length of turbulence scale.
Case No 3 Velocity is described by a set of functions with fulfillment of derivatives continuity.They are expressed by relationships: 0 015778 0 017937 0 292975 1 10 141232 Other parameters are calculated in the same way as in the case No 3.

Case No 4
Velocity is described as in the case No 3. Turbulence kinetic energy is described by a few polynomials with different degrees, which are determined on the basis of measurement results.They are written as: 22 759 The dissipation rate of kinetic turbulence energy is calculated from Eq. 7. Fig. 2 shows the profiles of input velocity and turbulence kinetic energy.The velocity graphs are similar, but the values of turbulence kinetic energy are quite different.The values for the case No 4 are close to measurements, but a physical relationship between flow parameters is not kept.It is caused by assumption functions for velocity and turbulence kinetic energy profiles which are in agreement with measurements.The errors in measurements cause the relation between input flow parameters not to be fully in agreement with equations of the turbulence model.

The analysis of flow parameters
The calculation and measurement results are shown in Figs 3, 4, 5, 6, 7 and 8.The figures contain the graphs of calculation results for all cases and for two k-ε methods: the standard one and RNG one.The calculation results are compared with our own measurement results.It arises from the research that: • the best results are obtained for the case No 4; • standard k-ε method is more sensitive to boundary conditions than RNG version; • differences between calculation results for all cases are bigger for turbulence kinetic energy than for velocity, but it is noted that the input profiles of turbulence kinetic energy vary more in the shapes and values than the graphs of velocities; • fulfillment of derivatives continuity has a positive influence on the quality of results, but the exact description of input flow parameter values is more important; • more exact description of the values of turbulence kinetic energy causes significant improvement of results for the u 2 components of velocity.Since the quality of calculation results depends on the correctness of the choice of model coefficients, the increase in sensitivity of flow parameters to these coefficients means the decrease in trust of obtained results.On the other side it should be noted that the parameters of the model are the factors of derivatives in a set of equations described in the k-ε model.When the term with a given parameter is large enough, then also the value of this parameter, being only a factor, has a greater influence on the solution.It is confirmed by the comparison of the turbulence kinetic energy graphs and its sensitivity to C ε1 .The fields of sensitivity coefficients show that fulfillment of derivatives continuity significantly decreases sensitivity of calculated flow parameters to the model coefficients and it means that it smooths away undesirable gradients of flow parameters functions.The increase in sensitivity is seen for the case No 4. It is probably caused by the lack of complete agreement between input flow parameters and equations of the k-ε model.

The analysis of sensitivity coefficients
where w -analysed flow parameter, ∆C m -increment of the C m parameter, w 1 -results of calculations at C C m m -D / 2 and w 2 -results of calculations at C C m m + D / 2.

Fig. 1 .
Fig. 1.The calculation domain with the description of boundary conditions.

Fig. 3 .
Fig. 3.The graphs of the component of velocity u 1 obtained from the standard k-ε method;  -measurement results, data for:  -case No 1,  -case No 2; s -case No 3,  -case No 4.

Figures 9
Figures 9 and 10 show model fields of sensitivity coefficients.Two coefficients are presented: the figures on the left side present sensitivity of the velocity components u 1 to the C ε1 coefficient and bitmaps on the right side show sensitivity of turbulence kinetic energy k to the C ε1 coefficient.The ranges of sensitivity coefficients are set at the same values in figures.The extreme values are different, but the figures are presented in such a way in order to make comparison areas of the same sensitivities.Since the quality of calculation results depends on the correctness of the choice of model coefficients, the increase in sensitivity of flow parameters to these coefficients means the decrease in trust of obtained results.On the other side it should be noted that the parameters of the model are the factors of derivatives in a set of equations described in the k-ε model.When the term with a given parameter

Fig. 9 .
Fig. 9. Model fields of sensitivity coefficients obtained using the standard version of k-ε method.

Fig. 10 .
Fig. 10.Model fields of sensitivity coefficients obtained using the RNG version of k-ε method.