LES of density ratio effects on film cooling under rotating frame

Abstract

A numerical study was conducted to investigate the performance of film cooling injection from a row of multiple square holes spaced laterally across a flat plate. LES with the standard Smagorinsky–Lilly model was used to investigate the dynamic mixing process between the coolant jet stream and the mainstream flows. The finite volume method and the unsteady PISO algorithm on a non-uniform staggered grid were applied. The values of rotation number (Ro) examined were 0.0, 0.03021, 0.06042, and 0.12084, jet spacing to jet width ratio (P/D) was 3.0, and a jet Reynolds number (Re) of 4700, which based on the hole width and the jet exit velocity. The effects of the coolant to mainstream density ratio (DR) on the film cooling effectiveness were investigated by injecting either nitrogen (DR = 0.98) or carbon dioxide (DR = 1.55) as the coolant streams. The effect of DR on the film-cooling effectiveness is coupled with varying velocity ratio (VR = 0.5 and 1.0). The coolant gas was injected at 90° to the mainstream flow. The flow fields of the present study were compared with experimental data in order to validate the reliability of the LES technique. It was shown that rotation has a strong impact on the jet trajectory behaviour and the film cooling effectiveness. In all cases, as the rotation number increases, the film effectiveness increases; this effect is increased as the velocity ratio increases. The results also showed the strong influence of velocity ratio on the flow field behaviour and the film cooling where the jet penetrates further into the cross flow as VR increases and the wake region increases with increasing VR. Furthermore, it was concluded that DR has only a minor effect on flow field and heat transfer at a constant velocity ratio.

Appendix: Synthesized turbulence

A time–space turbulent velocity field can be simulated by using random Fourier modes. This was proposed by Kraichnan [19] and Karweit et al. [18] and further developed by Bechara et al. [4] and Bailly and Juve [3]. The velocity field is given by the following equation:

$$u_{i}^{{\prime }} (x_{j} ) = 2\sum\limits_{m = 1}^{M} {\hat{u}^{m} } \cos \left( {k_{j}^{m} x_{j} + \psi^{m} } \right)\sigma_{i}^{m}$$

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The amplitude $\hat{u}^{m}$ of each mode can be calculated through the following equations:

$$\begin{aligned} & \hat{u}^{m} = \sqrt {E\left( {k_{j}^{m} } \right)\Delta k} \\ & E\left( {k_{j}^{m} } \right) = \alpha \frac{{u_{rms}^{2} }}{{k_{e} }}\frac{{\left( {k/k_{e} } \right)^{4} }}{{\left[ {1 + \left( {k/k_{e} } \right)^{2} } \right]^{{{{17} \mathord{\left/ {\vphantom {{17} 6}} \right. \kern-0pt} 6}}} }}e^{{\left[ { - 2\left( {k/k_{\eta } } \right)^{2} } \right]}} \\ & k = \left( {k_{i} k_{i} } \right)^{0.5} ,\quad k_{\eta } = \varepsilon^{0.25} \nu^{ - 0.75} ,\quad k_{e} = \alpha 9\pi /\left( {55L_{t} } \right)\quad {\text{where}}\,\alpha = 1.453 \\ \end{aligned}$$

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where $\hat{u}^{m}$, ψ ^m, and σ ^m _i are the amplitude, phase and direction of Fourier mode, respectively. u _rms represents the standard deviation of the streamwise velocity, ɛ is the rate of energy dissipation, ν is the kinematic viscosity and L _t is the turbulence length scale. In this study, the number of Fourier mode (m) is fixed to 150. The highest wave number is defined considering the mesh resolution, that is k _max = 2π/(2Δ), while k ₁ = k _e /p is defined as the smallest wave number. The factor p is chosen to be larger than one in order to make the largest scales larger than those corresponding to k _e . The notation used here follows that in [6, 31] and more information is given in these papers.

A fluctuating velocity field is generated each time step as described above. They are independent of each other, however, and their time correlation will thus be zero. This is unphysical. To create correlation in time, new fluctuating velocity fields, $(U_{i}^{{\prime }} )^{n} ,$ are computed based on an asymmetric time filter:

$$\left( {U_{i}^{{\prime }} } \right)^{n} = a\left( {U_{i}^{{\prime }} } \right)^{n - 1} + b\left( {u_{i}^{{\prime }} } \right)^{n}$$

where n is the time step number, a and b are constant coefficients taken as a = exp(−Δt/ι) and b = (1 − a ²)^0.5, respectively. And ι is the turbulence time scale, which is usually approximated as: ι = D/U _b . Consequently, the instantaneous velocity at the inflow is prescribed as:

$$\bar{u}_{i} (0,y,z,t) = U_{in(i)} (y) + u_{in(i)}^{{\prime }} (y,z,t)\quad {\text{where}}\,u_{in(i)}^{{\prime }} = \left( {U_{i}^{{\prime }} } \right)^{n}$$

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