Figures & data: Generalized UNIFAES derivatives applied to the correlations of the transport equations for fluctuating kinetic energy, helicity and enstrophy in an oscillated laminar flow

Numerical Heat Transfer, Part B: Fundamentals

An International Journal of Computation and Methodology

Volume 84, 2023 - Issue 3

243

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Figures & data

Figure 1. Control volume and coordinate notation in Finite Volume method.

Figure 2. Domain composed by perturbed field, measurements region and hyper-viscous field.

Table 1. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

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Table 2. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 9,600.

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Table 3. Statistics of the kinetic energy transport equations for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

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Table 4. Statistics of the transport equation for helicity for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

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Table 5. Statistics of the transport equation for enstrophy for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

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Table 6. Residuals of transport equations at finite meshes and extrapolations, Re = 300, node 4.

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Table 7. Extrapolated values of characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

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Figure 3. Evolution of scalar properties kinetic energy, modulus of helicity and enstrophy by linear generalized UNIFAES for varying Reynolds number in node 4.

Figure 4. Evolution of the relative magnitudes of the terms in the balance of the kinetic energy EquationEquation(43) $\frac{\partial U_{j} \bar{u_{i} u_{i}} / 2}{\partial x_{j}} - ν \frac{\partial^{2} {\bar{u_{i} u}}_{i} / 2}{\partial x_{j} \partial x_{j}} = - \bar{u_{i} u_{j}} S_{i j} - \frac{\partial \bar{u_{j} p}}{\partial x_{j}} - \frac{\partial \bar{u_{j} u_{i} u_{i}} / 2}{\partial x_{j}} - ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}}}$ (43) Equation(43(43) $\frac{\partial U_{j} \bar{u_{i} u_{i}} / 2}{\partial x_{j}} - ν \frac{\partial^{2} {\bar{u_{i} u}}_{i} / 2}{\partial x_{j} \partial x_{j}} = - \bar{u_{i} u_{j}} S_{i j} - \frac{\partial \bar{u_{j} p}}{\partial x_{j}} - \frac{\partial \bar{u_{j} u_{i} u_{i}} / 2}{\partial x_{j}} - ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}}}$ (43) Equation)(43) $\frac{\partial U_{j} \bar{u_{i} u_{i}} / 2}{\partial x_{j}} - ν \frac{\partial^{2} {\bar{u_{i} u}}_{i} / 2}{\partial x_{j} \partial x_{j}} = - \bar{u_{i} u_{j}} S_{i j} - \frac{\partial \bar{u_{j} p}}{\partial x_{j}} - \frac{\partial \bar{u_{j} u_{i} u_{i}} / 2}{\partial x_{j}} - ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}}}$ (43) for varying Reynolds number, node 4, $240 \times 60 \times 60$ mesh, linear generalized UNIFAES.

Figure 4. Evolution of the relative magnitudes of the terms in the balance of the kinetic energy EquationEquation(43) ∂Ujuiui¯/2∂xj−ν∂2uiu¯i/2∂xj∂xj=−uiuj¯Sij –∂ujp¯∂xj−∂ujuiui¯/2∂xj −ν∂ui∂xj∂ui∂xj¯(43) Equation(43(43) ∂Ujuiui¯/2∂xj−ν∂2uiu¯i/2∂xj∂xj=−uiuj¯Sij –∂ujp¯∂xj−∂ujuiui¯/2∂xj −ν∂ui∂xj∂ui∂xj¯(43) Equation)(43) ∂Ujuiui¯/2∂xj−ν∂2uiu¯i/2∂xj∂xj=−uiuj¯Sij –∂ujp¯∂xj−∂ujuiui¯/2∂xj −ν∂ui∂xj∂ui∂xj¯(43) for varying Reynolds number, node 4, 240×60×60 mesh, linear generalized UNIFAES.

Figure 5. Evolution of the relative magnitudes of the terms in the balance of the helicity EquationEquation(45) $\frac{\partial \bar{u_{i} w_{i}} U_{j}}{\partial x_{j}} - ν \frac{\partial^{2} \bar{{u_{i} w}_{i}}}{\partial x_{j} \partial x_{j}} = {\bar{\frac{\partial u_{i} u_{j}}{\partial x_{j}}} W}_{i} - \bar{u_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \bar{w_{i} \frac{\partial p}{\partial x_{i}}} + \bar{u_{i} w_{j} s_{i j}} - \frac{\partial \bar{u_{i} w_{i} u_{j}}}{\partial x_{j}} - 2 ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (45) Equation(45(45) $\frac{\partial \bar{u_{i} w_{i}} U_{j}}{\partial x_{j}} - ν \frac{\partial^{2} \bar{{u_{i} w}_{i}}}{\partial x_{j} \partial x_{j}} = {\bar{\frac{\partial u_{i} u_{j}}{\partial x_{j}}} W}_{i} - \bar{u_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \bar{w_{i} \frac{\partial p}{\partial x_{i}}} + \bar{u_{i} w_{j} s_{i j}} - \frac{\partial \bar{u_{i} w_{i} u_{j}}}{\partial x_{j}} - 2 ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (45) Equation)(45) $\frac{\partial \bar{u_{i} w_{i}} U_{j}}{\partial x_{j}} - ν \frac{\partial^{2} \bar{{u_{i} w}_{i}}}{\partial x_{j} \partial x_{j}} = {\bar{\frac{\partial u_{i} u_{j}}{\partial x_{j}}} W}_{i} - \bar{u_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \bar{w_{i} \frac{\partial p}{\partial x_{i}}} + \bar{u_{i} w_{j} s_{i j}} - \frac{\partial \bar{u_{i} w_{i} u_{j}}}{\partial x_{j}} - 2 ν \bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (45) for varying Reynolds number, node 4, $240 \times 60 \times 60$ mesh, linear generalized UNIFAES.

Figure 5. Evolution of the relative magnitudes of the terms in the balance of the helicity EquationEquation(45) ∂uiwi¯Uj∂xj−ν∂2uiwi¯∂xj∂xj=∂uiuj∂xj¯Wi−uiuj¯∂Wi∂xj−wi∂p∂xi¯+uiwjsij¯−∂uiwiuj¯∂xj−2ν∂ui∂xj∂wi∂xj¯(45) Equation(45(45) ∂uiwi¯Uj∂xj−ν∂2uiwi¯∂xj∂xj=∂uiuj∂xj¯Wi−uiuj¯∂Wi∂xj−wi∂p∂xi¯+uiwjsij¯−∂uiwiuj¯∂xj−2ν∂ui∂xj∂wi∂xj¯(45) Equation)(45) ∂uiwi¯Uj∂xj−ν∂2uiwi¯∂xj∂xj=∂uiuj∂xj¯Wi−uiuj¯∂Wi∂xj−wi∂p∂xi¯+uiwjsij¯−∂uiwiuj¯∂xj−2ν∂ui∂xj∂wi∂xj¯(45) for varying Reynolds number, node 4, 240×60×60 mesh, linear generalized UNIFAES.

Figure 6. Evolution of the relative magnitudes of the terms in the balance of the enstrophy EquationEquation(46) $\frac{\partial U_{j} \bar{w_{i} w_{i} / 2}}{\partial x_{j}} - υ \frac{\partial^{2} \bar{{w_{i} w}_{i}} / 2}{\partial x_{j} \partial x_{j}} = \bar{w_{i} s_{i j}} W_{j} + \bar{w_{i} w_{j}} S_{i j} - \bar{w_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \frac{\partial \bar{u_{j} w_{i} w_{i} / 2}}{\partial x_{j}} - \bar{w_{i} w_{j} s_{i j}} - υ \bar{\frac{\partial w_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (46) Equation(46(46) $\frac{\partial U_{j} \bar{w_{i} w_{i} / 2}}{\partial x_{j}} - υ \frac{\partial^{2} \bar{{w_{i} w}_{i}} / 2}{\partial x_{j} \partial x_{j}} = \bar{w_{i} s_{i j}} W_{j} + \bar{w_{i} w_{j}} S_{i j} - \bar{w_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \frac{\partial \bar{u_{j} w_{i} w_{i} / 2}}{\partial x_{j}} - \bar{w_{i} w_{j} s_{i j}} - υ \bar{\frac{\partial w_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (46) Equation)(46) $\frac{\partial U_{j} \bar{w_{i} w_{i} / 2}}{\partial x_{j}} - υ \frac{\partial^{2} \bar{{w_{i} w}_{i}} / 2}{\partial x_{j} \partial x_{j}} = \bar{w_{i} s_{i j}} W_{j} + \bar{w_{i} w_{j}} S_{i j} - \bar{w_{i} u_{j}} \frac{\partial W_{i}}{\partial x_{j}} - \frac{\partial \bar{u_{j} w_{i} w_{i} / 2}}{\partial x_{j}} - \bar{w_{i} w_{j} s_{i j}} - υ \bar{\frac{\partial w_{i}}{\partial x_{j}} \frac{\partial w_{i}}{\partial x_{j}}}$ (46) for varying Reynolds number, node 4, $240 \times 60 \times 60$ mesh, linear generalized UNIFAES.

Generalized UNIFAES derivatives applied to the correlations of the transport equations for fluctuating kinetic energy, helicity and enstrophy in an oscillated laminar flow

Table 1. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

Table 2. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 9,600.

Table 3. Statistics of the kinetic energy transport equations for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 4. Statistics of the transport equation for helicity for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 5. Statistics of the transport equation for enstrophy for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 6. Residuals of transport equations at finite meshes and extrapolations, Re = 300, node 4.

Table 7. Extrapolated values of characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

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Generalized UNIFAES derivatives applied to the correlations of the transport equations for fluctuating kinetic energy, helicity and enstrophy in an oscillated laminar flow

Figures & data

Table 1. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

Table 2. Characteristic fluctuation parameters in 18 regularly spaced points for Re = 9,600.

Table 3. Statistics of the kinetic energy transport equations for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 4. Statistics of the transport equation for helicity for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 5. Statistics of the transport equation for enstrophy for Re = 300 at node 4 with distinct algebraic forms, numerical schemes, and levels of refinement.

Table 6. Residuals of transport equations at finite meshes and extrapolations, Re = 300, node 4.

Table 7. Extrapolated values of characteristic fluctuation parameters in 18 regularly spaced points for Re = 300.

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