Equations and parameters - Revision history

Apwillis: /* Non-dimensionalisation / scales */

2024-11-05T14:52:21Z

Non-dimensionalisation / scales

← Older revision		Revision as of 07:52, 5 November 2024
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	The scales used are		The scales used are
	* $R$, the radius of the pipe.		* $R$, the radius of the pipe. ($R=D/2$, where $D$ is the diameter.)
	* $U_{cl}$, the centre-line velocity for laminar flow.		* $U_{cl}$, the centre-line velocity for laminar flow.
	* $R/U_{cl}$ for time.		* $R/U_{cl}$ for time.
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	* the non-dimensional radius is 1,		* the non-dimensional radius is 1,
	* the non-dimensional laminar flow is $W(r)=1-r^2$,		* the non-dimensional laminar flow is $W(r)=1-r^2$,
	* the bulk ~~velocity~~ is $\frac{1}{2}$.		* the non-dimensional bulk speed, when fixed, is $\frac{1}{2}$.

	Note that ~~$R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that~~ the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.		Note that when the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.

	The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.		The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.

	For 'lab-units', based on $D$ and $U_b$, 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$~~. See [[Table of unit conversions]]~~.		For 'lab-units', based on $D$ and $U_b$, see [[Table of unit conversions]].
			1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$.

	=== Dimensionless parameters ===		=== Dimensionless parameters ===

Apwillis: /* Non-dimensionalisation / scales */

2024-11-05T14:47:53Z

Non-dimensionalisation / scales

← Older revision		Revision as of 07:47, 5 November 2024
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	In computational units,		In computational units,
	* the non-dimensional radius is 1 ~~and~~		* the non-dimensional radius is 1,
	* the non-dimensional laminar flow is $W(r)=1-r^2$.		* the non-dimensional laminar flow is $W(r)=1-r^2$,
			* the bulk velocity is $\frac{1}{2}$.

	Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.		Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.
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	The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.		The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.

	For 'lab-units', based on $D$ and $U_b$,		For 'lab-units', based on $D$ and $U_b$, 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$. See [[Table of unit conversions]].
	* 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$,
	* the bulk velocity corresponds to $\frac{1}{2}$ in code units.
	[[Table of unit conversions]].

	=== Dimensionless parameters ===		=== Dimensionless parameters ===

Apwillis: /* Non-dimensionalisation / scales */

2024-11-05T14:45:31Z

Non-dimensionalisation / scales

← Older revision		Revision as of 07:45, 5 November 2024
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	Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.		Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.

			The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.

	For 'lab-units', based on $D$ and $U_b$,		For 'lab-units', based on $D$ and $U_b$,

Apwillis: /* Dimensionless parameters */

2021-07-05T14:19:50Z

Dimensionless parameters

← Older revision		Revision as of 07:19, 5 July 2021
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	=== Dimensionless parameters ===		=== Dimensionless parameters ===

	Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu = DU_b / \nu$, where $\nu = \mu / \rho$.		Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu = DU_b / \nu$, where the kinematic viscosity $\nu = \mu / \rho$.

	Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.		Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.

Apwillis: /* Dimensionless parameters */

2021-07-05T14:18:23Z

Dimensionless parameters

← Older revision		Revision as of 07:18, 5 July 2021
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	=== Dimensionless parameters ===		=== Dimensionless parameters ===

	Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu = DU_b\~~rho~~/ \mu$.		Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu = DU_b / \nu$, where $\nu = \mu / \rho$.

	Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.		Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.

Apwillis: /* Dimensionless parameters */

2020-10-19T09:18:39Z

Dimensionless parameters

← Older revision		Revision as of 02:18, 19 October 2020
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	For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.		For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.

	$1+\beta = Re / Re_m$ is an observed quantity. For fixed flux, $1+\beta=\langle\partial p/\partial z\rangle \,/\, (dP/dz)$, where $(dP/dz)$ is the laminar pressure gradient and $\langle\partial p/\partial z\rangle$ is the average pressure gradient observed. For fixed pressure, $1+\beta=U_{cl}/~~U_b~~$, where $U_b$ is the observed bulk speed.		$1+\beta = Re / Re_m$ is an observed quantity. For fixed flux, $1+\beta=\langle\partial p/\partial z\rangle \,/\, (dP/dz)$, where $(dP/dz)$ is the laminar pressure gradient and $\langle\partial p/\partial z\rangle$ is the average pressure gradient observed. For fixed pressure, $1+\beta=U_{cl}/(2U_b)$, where $U_b$ is the observed bulk speed.

	The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [https://en.wikipedia.org/wiki/Shear_velocity],		The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [https://en.wikipedia.org/wiki/Shear_velocity],

Apwillis: /* Dimensionless parameters */

2020-10-19T09:15:32Z

Dimensionless parameters

← Older revision		Revision as of 02:15, 19 October 2020
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	For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.		For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.

	$1+\beta = Re / Re_m$ is an observed quantity. (For fixed flux, $Re$ is ~~inferred from~~ the ~~observed~~ pressure gradient and ~~its corresponding~~ $~~U_{cl}~~$. For fixed pressure ~~gradient~~, $~~Re_m~~$ is ~~inferred from~~ the observed ~~$U_b$~~.)		$1+\beta = Re / Re_m$ is an observed quantity. For fixed flux, $1+\beta=\langle\partial p/\partial z\rangle \,/\, (dP/dz)$, where $(dP/dz)$ is the laminar pressure gradient and $\langle\partial p/\partial z\rangle$ is the average pressure gradient observed. For fixed pressure, $1+\beta=U_{cl}/U_b$, where $U_b$ is the observed bulk speed.

	The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [https://en.wikipedia.org/wiki/Shear_velocity],		The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [https://en.wikipedia.org/wiki/Shear_velocity],

Apwillis: /* Dimensionless parameters */

2020-10-16T10:28:51Z

Dimensionless parameters

← Older revision		Revision as of 03:28, 16 October 2020
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	Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.		Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.

	For fixed flow rate, $U_{cl}=2\,U_b$ at all times. The Reynolds number $Re_m$ is more commonly defined in terms of the constant mean speed $U_b$.		For fixed flux (constant flow rate), $U_{cl}=2\,U_b$ at all times. The Reynolds number $Re_m$ is more commonly defined in terms of the constant mean speed $U_b$.

	For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.		For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.

Apwillis: /* Dimensionless parameters */

2020-10-16T10:26:09Z

Dimensionless parameters

← Older revision		Revision as of 03:26, 16 October 2020
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	Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.		Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.

	For fixed ~~flux~~ $1+\beta = Re / Re_m$ is an observed quantity, and		For fixed flow rate, $U_{cl}=2\,U_b$ at all times. The Reynolds number $Re_m$ is more commonly defined in terms of the constant mean speed $U_b$.
	$Re_\tau=u_\tau R/\nu = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$
			For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.

			$1+\beta = Re / Re_m$ is an observed quantity. (For fixed flux, $Re$ is inferred from the observed pressure gradient and its corresponding $U_{cl}$. For fixed pressure gradient, $Re_m$ is inferred from the observed $U_b$.)

			The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [https://en.wikipedia.org/wiki/Shear_velocity],
			is given by $Re_\tau = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$.

	=== Evolution equations ===		=== Evolution equations ===

Apwillis: /* Non-dimensionalisation / scales */

2020-10-16T10:13:20Z

Non-dimensionalisation / scales

← Older revision		Revision as of 03:13, 16 October 2020
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	Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.		Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.

	For 'lab-units', based on $D$ and $U_b$,		For 'lab-units', based on $D$ and $U_b$,
	* 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$,		* 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$,