Planetesimal-Erosion - Revision history

Emilio.S: /* Resolution study */

2025-09-25T09:37:26Z

Resolution study

← Older revision		Revision as of 11:37, 25 September 2025
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	It is to be expected that the simulation data and, accordingly, the results for very coarse grids may be distorted due to the discretisation of the space. It is therefore advisable to first carry out several simulations for grids with different resolutions. The aim is to find the coarsest grid for which the relevant quantities, such as normal stress and shear stress, no longer change or change only slightly. For this purpose, parameters such as the Mach number and the Reynolds number are kept constant in the corresponding simulations.		It is to be expected that the simulation data and, accordingly, the results for very coarse grids may be distorted due to the discretisation of the space. It is therefore advisable to first carry out several simulations for grids with different resolutions. The aim is to find the coarsest grid for which the relevant quantities, such as normal stress and shear stress, no longer change or change only slightly. For this purpose, parameters such as the Mach number and the Reynolds number are kept constant in the corresponding simulations.

	First, the parameters are selected in line with the bachelor's thesis [https://wiki.uni-due.de/agk/index.php?title=BA_Emilio_Schmidt (Emilio Schmidt, 2024)]. Specifically, a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\) and a Reynolds number of \(\text{Re} = 10^{-2}\) are selected. This choice serves to verify whether the transition to the new system of natural units has been consistent. In particular, it should be ensured that the relevant quantities also converge in this system of units with increasing resolution.		First, the parameters are selected in line with the bachelor's thesis ([https://wiki.uni-due.de/agk/index.php?title=BA_Emilio_Schmidt Emilio Schmidt, 2024]). Specifically, a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\) and a Reynolds number of \(\text{Re} = 10^{-2}\) are selected. This choice serves to verify whether the transition to the new system of natural units has been consistent. In particular, it should be ensured that the relevant quantities also converge in this system of units with increasing resolution.

	For the first simulation of this resolution study, a grid was selected whose inner radius \(R_I/R=1\) corresponds to the radius of the sphere being flowed upon in the selected natural units and whose outer radius is \(R_A/R=100\). The grid initially contains \(22\) cells in the radial direction, which increase linearly with the outer radius. In the polar direction, there are \(15\) cells of uniform size. In the following simulations, the number of cells in the radial and polar directions is multiplied by a natural number \(n\in\mathbb{N}\). This results in a cell count of \(n\cdot 22\) in the radial direction and \(n\cdot 15\) in the polar direction. In the following, such a grid is referred to as a grid with resolution \(n\).		For the first simulation of this resolution study, a grid was selected whose inner radius \(R_I/R=1\) corresponds to the radius of the sphere being flowed upon in the selected natural units and whose outer radius is \(R_A/R=100\). The grid initially contains \(22\) cells in the radial direction, which increase linearly with the outer radius. In the polar direction, there are \(15\) cells of uniform size. In the following simulations, the number of cells in the radial and polar directions is multiplied by a natural number \(n\in\mathbb{N}\). This results in a cell count of \(n\cdot 22\) in the radial direction and \(n\cdot 15\) in the polar direction. In the following, such a grid is referred to as a grid with resolution \(n\).

Emilio.S: /* Resolution study */ Picture

2025-09-25T09:26:58Z

Resolution study: Picture

← Older revision		Revision as of 11:26, 25 September 2025
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	The following figure shows both the frontal normal stress, i.e. the normal stress acting on the sphere in the direction of flow, and the frontal shear stress across the grid resolution, including a power law fit:		The following figure shows both the frontal normal stress, i.e. the normal stress acting on the sphere in the direction of flow, and the frontal shear stress across the grid resolution, including a power law fit:

	[[File:Resolution study.png\|thumb\|Resolution study for a Reynolds number of \(\text{Re} = 10^{2}\) and a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\). The values of the frontal normal stress (left) and the frontal shear stress (right) are shown as a function of the grid resolution \(n\). Both variables were additionally fitted with a power law. The results show that both normal and shear stress converge with increasing resolution.]]		[[File:Resolution study.png\|750px\|thumb\|center\|Resolution study for a Reynolds number of \(\text{Re} = 10^{-2}\) and a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\). The values of the frontal normal stress (left) and the frontal shear stress (right) are shown as a function of the grid resolution \(n\). Both variables were additionally fitted with a power law. The results show that both normal and shear stress converge with increasing resolution.]]

	~~PICTURE~~

	The figure shows that the two quantities converge as the resolution increases. This confirms that the transition to the new system of natural units has been consistent.		The figure shows that the two quantities converge as the resolution increases. This confirms that the transition to the new system of natural units has been consistent.
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	The following figure shows both the frontal normal stress and the frontal shear stress across the grid resolution, including the power law fit:		The following figure shows both the frontal normal stress and the frontal shear stress across the grid resolution, including the power law fit:

	~~PICTURE~~		[[File:Resolution study RE1E+3.png\|750px\|thumb\|center\|Resolution study for a Reynolds number of \(\text{Re} = 10^{3}\) and a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\). The values of the frontal normal stress (left) and the frontal shear stress (right) are shown as a function of the grid resolution. Both variables were additionally fitted with a power law. The results show that both normal and shear stress converge with increasing resolution.]]

	The figure shows that both quantities converge with increasing grid resolution even at an elevated Reynolds number of \(\text{Re} = 10^{3}\). However, it is noticeable that convergence is significantly slower than in the case of the smaller Reynolds number \(\text{Re} = 10^{-2}\). In addition, the frontal normal stress exhibits different behaviour. While it decreases or increases in magnitude with increasing resolution at \(\text{Re} = 10^{3}\), it behaves in the opposite manner in the case of \(\text{Re} = 10^{-2}\), increasing or decreasing in magnitude with increasing resolution.		The figure shows that both quantities converge with increasing grid resolution even at an elevated Reynolds number of \(\text{Re} = 10^{3}\). However, it is noticeable that convergence is significantly slower than in the case of the smaller Reynolds number \(\text{Re} = 10^{-2}\). In addition, the frontal normal stress exhibits different behaviour. While it decreases or increases in magnitude with increasing resolution at \(\text{Re} = 10^{3}\), it behaves in the opposite manner in the case of \(\text{Re} = 10^{-2}\), increasing or decreasing in magnitude with increasing resolution.

Emilio.S: /* Resolution study */ Picture

2025-09-25T09:07:36Z

Resolution study: Picture

← Older revision		Revision as of 11:07, 25 September 2025
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	The following figure shows both the frontal normal stress, i.e. the normal stress acting on the sphere in the direction of flow, and the frontal shear stress across the grid resolution, including a power law fit:		The following figure shows both the frontal normal stress, i.e. the normal stress acting on the sphere in the direction of flow, and the frontal shear stress across the grid resolution, including a power law fit:

			[[File:Resolution study.png\|thumb\|Resolution study for a Reynolds number of \(\text{Re} = 10^{2}\) and a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\). The values of the frontal normal stress (left) and the frontal shear stress (right) are shown as a function of the grid resolution \(n\). Both variables were additionally fitted with a power law. The results show that both normal and shear stress converge with increasing resolution.]]

	PICTURE		PICTURE

Emilio.S: /* Resolution study */

2025-09-24T20:49:57Z

Resolution study

← Older revision		Revision as of 22:49, 24 September 2025
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	The different convergence behaviour can be explained by the fact that the viscosity term, or more precisely the shear viscosity term, scales inversely with the Reynolds number in the normal stress in the selected system of natural units. For large Reynolds numbers, this term is negligible, so that the normal stress on the sphere surface is essentially determined by the negative pressure. As the resolution increases, the cell centres of the cells surrounding the particle move closer to the sphere surface, reducing the error in pressure extrapolation. At coarse resolutions, the pressure is systematically underestimated because the extrapolation is performed from cell centres that are further away from the surface. At low Reynolds numbers, however, the viscosity term cannot be neglected, but rather becomes dominant. Since the fluid is slowed down when it hits the sphere, the radial derivative of the radial velocity component on the sphere surface is positive and thus contributes positively to the normal stress. This derivative increases with increasing resolution for the same reason that the pressure increases at high Reynolds numbers. The cell centres closer to the surface provide more accurate values. Due to the dominance of the viscosity term, this contribution outweighs the increase in pressure, so that the normal stress increases with increasing resolution at low Reynolds numbers or decreases in magnitude.		The different convergence behaviour can be explained by the fact that the viscosity term, or more precisely the shear viscosity term, scales inversely with the Reynolds number in the normal stress in the selected system of natural units. For large Reynolds numbers, this term is negligible, so that the normal stress on the sphere surface is essentially determined by the negative pressure. As the resolution increases, the cell centres of the cells surrounding the particle move closer to the sphere surface, reducing the error in pressure extrapolation. At coarse resolutions, the pressure is systematically underestimated because the extrapolation is performed from cell centres that are further away from the surface. At low Reynolds numbers, however, the viscosity term cannot be neglected, but rather becomes dominant. Since the fluid is slowed down when it hits the sphere, the radial derivative of the radial velocity component on the sphere surface is positive and thus contributes positively to the normal stress. This derivative increases with increasing resolution for the same reason that the pressure increases at high Reynolds numbers. The cell centres closer to the surface provide more accurate values. Due to the dominance of the viscosity term, this contribution outweighs the increase in pressure, so that the normal stress increases with increasing resolution at low Reynolds numbers or decreases in magnitude.


	To be continued...		To be continued...

Emilio.S: Results and discussion/Pictures will be provided later

2025-09-24T20:49:35Z

Results and discussion/Pictures will be provided later

← Older revision		Revision as of 22:49, 24 September 2025
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	This means that the stresses for each cell in the simulation area, and in particular at its inner edge, can now be determined with second-order accuracy.		This means that the stresses for each cell in the simulation area, and in particular at its inner edge, can now be determined with second-order accuracy.

			= Results and discussion =

			Before the simulations for tuning the Mach numbers can be carried out, a suitable setup must first be found. This setup should be chosen in such a way that, on the one hand, the results are not distorted by the choice of setup and, on the other hand, the simulation duration remains within reasonable limits.

			== Resolution study ==

			It is to be expected that the simulation data and, accordingly, the results for very coarse grids may be distorted due to the discretisation of the space. It is therefore advisable to first carry out several simulations for grids with different resolutions. The aim is to find the coarsest grid for which the relevant quantities, such as normal stress and shear stress, no longer change or change only slightly. For this purpose, parameters such as the Mach number and the Reynolds number are kept constant in the corresponding simulations.

			First, the parameters are selected in line with the bachelor's thesis [https://wiki.uni-due.de/agk/index.php?title=BA_Emilio_Schmidt (Emilio Schmidt, 2024)]. Specifically, a Mach number of \(\mathcal{M} = 5 \cdot 10^{-3}\) and a Reynolds number of \(\text{Re} = 10^{-2}\) are selected. This choice serves to verify whether the transition to the new system of natural units has been consistent. In particular, it should be ensured that the relevant quantities also converge in this system of units with increasing resolution.

			For the first simulation of this resolution study, a grid was selected whose inner radius \(R_I/R=1\) corresponds to the radius of the sphere being flowed upon in the selected natural units and whose outer radius is \(R_A/R=100\). The grid initially contains \(22\) cells in the radial direction, which increase linearly with the outer radius. In the polar direction, there are \(15\) cells of uniform size. In the following simulations, the number of cells in the radial and polar directions is multiplied by a natural number \(n\in\mathbb{N}\). This results in a cell count of \(n\cdot 22\) in the radial direction and \(n\cdot 15\) in the polar direction. In the following, such a grid is referred to as a grid with resolution \(n\).

			The following figure shows both the frontal normal stress, i.e. the normal stress acting on the sphere in the direction of flow, and the frontal shear stress across the grid resolution, including a power law fit:

			PICTURE

			The figure shows that the two quantities converge as the resolution increases. This confirms that the transition to the new system of natural units has been consistent.

			In the next step, an analogue resolution study is carried out, but for Reynolds numbers as used in [https://iopscience.iop.org/article/10.3847/1538-4357/ac1e88/meta Cedenblad et al. (2021)]. The same grid and Mach number are used, but the Reynolds number is now set to \(\text{Re} = 10^{3}\) instead of \(\text{Re} = 10^{-2}\).

			The following figure shows both the frontal normal stress and the frontal shear stress across the grid resolution, including the power law fit:

			PICTURE

			The figure shows that both quantities converge with increasing grid resolution even at an elevated Reynolds number of \(\text{Re} = 10^{3}\). However, it is noticeable that convergence is significantly slower than in the case of the smaller Reynolds number \(\text{Re} = 10^{-2}\). In addition, the frontal normal stress exhibits different behaviour. While it decreases or increases in magnitude with increasing resolution at \(\text{Re} = 10^{3}\), it behaves in the opposite manner in the case of \(\text{Re} = 10^{-2}\), increasing or decreasing in magnitude with increasing resolution.

			The slow convergence can be explained by the behaviour of the boundary layer thickness \(\delta\), which generally scales with the Reynolds number for both laminar and turbulent flows as follows (see [https://en.wikipedia.org/wiki/Boundary_layer_thickness Wikipedia/Boundary_layer_thickness]):

			\begin{align}
			\delta \propto \text{Re}^{-a}, \quad a \in (0,1).
			\end{align}

			The exponent \(a\) takes on different values in the interval \((0,1)\) depending on the flow regime, but this is not relevant to the present argument. The decisive factor is that the boundary layer thickness decreases with increasing Reynolds number. Since the distributions of density, pressure and velocity within the boundary layer contribute significantly to the calculation of the stresses under consideration, this must be resolved numerically with sufficient accuracy. Consequently, higher Reynolds numbers require a correspondingly higher grid resolution in order to adequately represent the thinner boundary layer.

			The different convergence behaviour can be explained by the fact that the viscosity term, or more precisely the shear viscosity term, scales inversely with the Reynolds number in the normal stress in the selected system of natural units. For large Reynolds numbers, this term is negligible, so that the normal stress on the sphere surface is essentially determined by the negative pressure. As the resolution increases, the cell centres of the cells surrounding the particle move closer to the sphere surface, reducing the error in pressure extrapolation. At coarse resolutions, the pressure is systematically underestimated because the extrapolation is performed from cell centres that are further away from the surface. At low Reynolds numbers, however, the viscosity term cannot be neglected, but rather becomes dominant. Since the fluid is slowed down when it hits the sphere, the radial derivative of the radial velocity component on the sphere surface is positive and thus contributes positively to the normal stress. This derivative increases with increasing resolution for the same reason that the pressure increases at high Reynolds numbers. The cell centres closer to the surface provide more accurate values. Due to the dominance of the viscosity term, this contribution outweighs the increase in pressure, so that the normal stress increases with increasing resolution at low Reynolds numbers or decreases in magnitude.


			To be continued...

Emilio.S: /* Numerical methods */ Numerical calculation of the stresses

2025-07-01T09:05:35Z

Numerical methods: Numerical calculation of the stresses

← Older revision		Revision as of 11:05, 1 July 2025
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	Due to the fact that a two-dimensional computational domain is considered because of the axial symmetry of the system, the computational domain has the shape of a half disc with a hole in the origin, of diameter \(2R\). Therefore, boundary conditions must also be specified for the axes \((r,\theta=0)\) and \((r,\theta=\pi)\). The so-called axissymmetric boundary condition is applied there. This boundary condition ensures axial symmetry in the simulations. A more detailed explanation of this type of boundary condition can also be found in the [https://plutocode.ph.unito.it/userguide.pdf PLUTO User's Guide].		Due to the fact that a two-dimensional computational domain is considered because of the axial symmetry of the system, the computational domain has the shape of a half disc with a hole in the origin, of diameter \(2R\). Therefore, boundary conditions must also be specified for the axes \((r,\theta=0)\) and \((r,\theta=\pi)\). The so-called axissymmetric boundary condition is applied there. This boundary condition ensures axial symmetry in the simulations. A more detailed explanation of this type of boundary condition can also be found in the [https://plutocode.ph.unito.it/userguide.pdf PLUTO User's Guide].

			== Numerical calculation of the stresses ==

			As PLUTO only outputs the data for the respective cell centres, the pressure and velocity fields in particular are not output on the surface of the sphere. However, these fields are required to calculate the stresses acting on the sphere.

			The velocity field on the surface of the sphere is already determined by the boundary conditions. However, the pressure on the surface of the sphere is unknown and must therefore be extrapolated. Linear extrapolation is suitable here, as it has a second-order error which corresponds to the errors of the methods used to determine the derivatives of the fields. For the linear extrapolation of the pressure on the spherical surface \(p(r=R)=p_0\), the next two data points \((r_1,p_1)\) and \((r_2,p_2)\) are required. The extrapolated pressure is then given by:

			\begin{align}
			p_0=p_2+\frac{R-r_2}{r_1-r_2}(p_1-p_2)
			\end{align}

			The derivatives are determined numerically using the Numpy function <code>numpy.gradient</code>. This function can approximate derivatives of functions defined by discrete data points and has a second-order error. A key advantage of this function is its ability to calculate derivatives with inhomogeneous step sizes \(\Delta r_i\). This property is of particular importance in the context of this project, as the cell size in the simulation grid under consideration increases linearly in the radial direction. Correspondingly, the distances between the output data points and thus the step sizes also increase linearly.

			For the present investigation, mainly derivatives at the inner edge of the simulation area are required. By default, <code>numpy.gradient</code> uses one-sided first-order differences at the edges, which results in a first-order error there. By setting the parameter <code>edge_order=2</code>, one-sided second-order differences are used instead, so that a second-order error is also guaranteed at the boundary points. A detailed explanation of the exact calculation of the derivatives can be found in the [https://numpy.org/doc/2.1/reference/generated/numpy.gradient.html Numpy documentation].

			This means that the stresses for each cell in the simulation area, and in particular at its inner edge, can now be determined with second-order accuracy.

Emilio.S: /* Natural units */

2025-06-24T21:14:58Z

Natural units

← Older revision		Revision as of 23:14, 24 June 2025
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	\begin{align}		\begin{align}
	&\sigma^\Big\|_{r^=1} = - p^\Big\|_{r^=1} + \frac{4}{3\text{Re}_{\infty}}		&\sigma^\Big\|_{r^=1} = - p^\Big\|_{r^=1} + \frac{8}{3\text{Re}_{\infty}}
	\frac{\partial u_r^}{\partial r^}\Big\|_{r^*=1}		\frac{\partial u_r^}{\partial r^}\Big\|_{r^*=1}
	\\[6pt]		\\[6pt]
	&\tau^\Big\|_{r^=1} =		&\tau^\Big\|_{r^=1} =
	\frac{1}{\text{Re}_{\infty}}\bigg\|\frac{\partial u_\theta^}{\partial r^}\Big\|_{r^*=1}\bigg\|		\frac{2}{\text{Re}_{\infty}}\bigg\|\frac{\partial u_\theta^}{\partial r^}\Big\|_{r^*=1}\bigg\|
	\end{align}		\end{align}

Emilio.S: /* Natural units */

2025-06-24T21:14:22Z

Natural units

← Older revision		Revision as of 23:14, 24 June 2025
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	\\[6pt]		\\[6pt]
	\Leftrightarrow\;&		\Leftrightarrow\;&
	\mu_{\infty}^* = \frac{1}{\text{Re}_{\infty}}		\mu_{\infty}^* = \frac{2}{\text{Re}_{\infty}}
	\end{align}		\end{align}

Emilio.S: /* Natural units */

2025-06-24T21:14:09Z

Natural units

← Older revision		Revision as of 23:14, 24 June 2025
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	\begin{align}		\begin{align}
	\phantom{\Rightarrow}\;&		\phantom{\Rightarrow}\;&
	\mu_{\infty}^* = \frac{\mu_{\infty}}{\rho_{\infty} u_{\infty} R} \; , \quad \mu_{\infty} = \frac{\rho_{\infty} u_{\infty} R}{\text{Re}_{\infty}}		\mu_{\infty}^* = \frac{\mu_{\infty}}{\rho_{\infty} u_{\infty} R} \; , \quad
			\mu_{\infty} = \frac{2\rho_{\infty} u_{\infty} R}{\text{Re}_{\infty}}
	\\[6pt]		\\[6pt]
	\Leftrightarrow\;&		\Leftrightarrow\;&

Emilio.S: /* Computational domain */

2025-06-24T09:46:30Z

Computational domain

@@ Line 356: / Line 356: @@
 == Computational domain ==
-As explained a sphere with the radius \(R\) around which a viscous fluid flows with an inflow velocity \(\boldsymbol{u}=-u_{\infty}\boldsymbol{e}_z\) is investigated. Due to the spherical symmetry of the problem, the computational domain is modelled in spherical coordinates, with the centre of the sphere at the origin of the coordinate system. However, due to the axial symmetry, it is sufficient to set up a two-dimensional simulation domain, whereby only the radial component \(r\) and the polar component \(\theta\) are considered.
+As explained a sphere with the radius \(R\) around which a viscous fluid flows with an inflow velocity \(\boldsymbol{u}_{\infty}=-u_{\infty}\boldsymbol{e}_z\) is investigated. Due to the spherical symmetry of the problem, the computational domain is modelled in spherical coordinates, with the centre of the sphere at the origin of the coordinate system. However, due to the axial symmetry, it is sufficient to set up a two-dimensional simulation domain, whereby only the radial component \(r\) and the polar component \(\theta\) are considered.
 The grid extends from an inner radius \(R_{\text{I}}\), which corresponds to the spherical radius \(R\), to an outer radius \(R_{\text{A}}\). In the polar direction, the grid starts at an angle of \(\theta_\text{S}=0\) and extends to \(\theta_{\text{E}}=\pi\). The number of cells \(N_r\) in the radial direction and \(N_\theta\) in the polar direction is selected so that the cells have an approximately square shape. This configuration is crucial for the numerical precision of the simulations, as highly distorted cells are more susceptible to numerical errors.
@@ Line 366: / Line 366: @@
 \big(10^{\Delta\xi}-1\big)
 \end{align}