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ABSTRACT
In this work, with elasticity governed by the Giesekus constitutive equation, a proper orthogonal decomposition (POD) reduced-order model of viscoelastic turbulent channel flow is established for the first time. The established reduced-order model is based on small sets of basis functions from the POD of the sampling data obtained by direct numerical simulation (DNS) for the studied flow. The POD reduced-order model is tested on cases with that are different from the samplings for viscoelastic turbulent channel flow. The results show that the errors for root-mean-square (rms) velocity fluctuations are significant at the top and bottom walls. It is found that each basis function plays an important role in describing the studied turbulence which makes it unmanageable to obtain accurate velocity field (including mean velocity and velocity fluctuations) through solving the reduced-order model. It is of necessity to take all the basis functions into consideration to depict the flows more accurately. However, the mean velocity obtained from the reduced-order model is of high precision, which states that the POD-based reduced-order model is a potential approach to obtain an accurate mean velocity field for viscoelastic turbulent flow, which has great significance in academic study as well as engineering. The calculation speed of the established reduced-order model is much faster than that of DNS, which indicates that the POD is a highly efficient way of obtaining the statistic characteristics, such as mean velocity in turbulent channel flow.
Nomenclature
| Roman symbols | = | |
| ak | = | spectral coefficient of velocity |
| bk | = | spectral coefficient of deformation rate |
| c | = | deformation rate |
| = | mean deformation rate | |
| = | fluctuation of deformation rate | |
| h | = | half of the channel height, m |
| M1 | = | total number of the velocity basis functions chosen in the model |
| M2 | = | total number of the deformation rate basis functions chosen in the model |
| N | = | total sampling number |
| p | = | pressure, Pa |
| = | Reynolds number | |
| t | = | time, s |
| u | = | velocity vector, m/s |
| ū | = | mean velocity vector, m/s |
| = | fluctuation of velocity, m/s | |
| = | friction velocity, m/s | |
| = | Weissenberg number | |
| x | = | unit vector in space, m |
| = | coefficients of the momentum reduced-order model for viscoelastic turbulent channel flow | |
| = | coefficients of the constitutive reduced-order model for viscoelastic turbulent channel flow | |
| Greek symbols | = | |
| α | = | liquidity factor |
| β | = | ratio of viscosity for solvent to zero shear viscosity for solution, |
| = | Kronecker delta | |
| η | = | zero shear viscosity, Pa.s |
| λ | = | relaxation time, s |
| μ | = | dynamic viscosity, Pa.s |
| ρ | = | density, kg/m3 |
| = | viscous stress of the viscoelastic fluid, Pa | |
| Φ | = | velocity basis function |
| φ | = | deformation rate basis function |
| = | Eigenvalue | |
| = | energy contribution | |
| = | accumulative energy contribution |
Nomenclature
| Roman symbols | = | |
| ak | = | spectral coefficient of velocity |
| bk | = | spectral coefficient of deformation rate |
| c | = | deformation rate |
| = | mean deformation rate | |
| = | fluctuation of deformation rate | |
| h | = | half of the channel height, m |
| M1 | = | total number of the velocity basis functions chosen in the model |
| M2 | = | total number of the deformation rate basis functions chosen in the model |
| N | = | total sampling number |
| p | = | pressure, Pa |
| = | Reynolds number | |
| t | = | time, s |
| u | = | velocity vector, m/s |
| ū | = | mean velocity vector, m/s |
| = | fluctuation of velocity, m/s | |
| = | friction velocity, m/s | |
| = | Weissenberg number | |
| x | = | unit vector in space, m |
| = | coefficients of the momentum reduced-order model for viscoelastic turbulent channel flow | |
| = | coefficients of the constitutive reduced-order model for viscoelastic turbulent channel flow | |
| Greek symbols | = | |
| α | = | liquidity factor |
| β | = | ratio of viscosity for solvent to zero shear viscosity for solution, |
| = | Kronecker delta | |
| η | = | zero shear viscosity, Pa.s |
| λ | = | relaxation time, s |
| μ | = | dynamic viscosity, Pa.s |
| ρ | = | density, kg/m3 |
| = | viscous stress of the viscoelastic fluid, Pa | |
| Φ | = | velocity basis function |
| φ | = | deformation rate basis function |
| = | Eigenvalue | |
| = | energy contribution | |
| = | accumulative energy contribution |