Cebeci-Smith model - Revision history

Peter: Reverted edits by RicgeTcnac (Talk) to last version by Merrifj

2008-12-18T12:13:12Z

Reverted edits by RicgeTcnac (Talk) to last version by Merrifj

← Older revision		Revision as of 12:13, 18 December 2008
Line 1:		Line 1:
-	~~letoaloucotr~~
	{{Turbulence modeling}}		{{Turbulence modeling}}
	The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.		The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.

RicgeTcnac: labaserdrond

2008-12-18T12:06:59Z

labaserdrond

← Older revision		Revision as of 12:06, 18 December 2008
Line 1:		Line 1:
		+	letoaloucotr
	{{Turbulence modeling}}		{{Turbulence modeling}}
	The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.		The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.

Merrifj: /* Equations */

2008-06-11T15:18:33Z

Equations

← Older revision		Revision as of 15:18, 11 June 2008
Line 24:		Line 24:
	<table width="70%"><tr><td>		<table width="70%"><tr><td>
	:<math>		:<math>
-	{\mu_t}_{inner} = \rho l^2 l \left[\left(	+	{\mu_t}_{inner} = \rho l^2 \left[\left(
	\frac{\partial U}{\partial y}\right)^2 +		\frac{\partial U}{\partial y}\right)^2 +
	\left(\frac{\partial V}{\partial x}\right)^2		\left(\frac{\partial V}{\partial x}\right)^2
Line 51:		Line 51:
	</math></td><td width="5%">(6)</td></tr></table>		</math></td><td width="5%">(6)</td></tr></table>

-	where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by	+	where <math>\alpha=0.0168</math> and <math>\delta_v^*</math> is the velocity thickness given by

	<table width="70%"><tr><td>		<table width="70%"><tr><td>
	:<math>		:<math>
-	\delta_v^* = \int_0^\delta (1-U/U_e)dy,	+	\delta_v^* = \int_0^\delta (1-U/U_e)dy.
	</math></td><td width="5%">(7)</td></tr></table>		</math></td><td width="5%">(7)</td></tr></table>

-	~~and~~ <math>F_{KLEB}</math> is the Klebanoff intermittency function given by	+	<math>F_{KLEB}</math> is the Klebanoff intermittency function given by

	<table width="100%"><tr><td>		<table width="100%"><tr><td>

Shweta: /* Equations */

2007-09-17T17:23:28Z

Equations

← Older revision		Revision as of 17:23, 17 September 2007
Line 65:		Line 65:
	\right]^{-1}		\right]^{-1}
	</math></td><td width="5%">(8)</td></tr></table>		</math></td><td width="5%">(8)</td></tr></table>
-

	== Model variants ==		== Model variants ==

Jola at 09:41, 12 June 2007

2007-06-12T09:41:22Z

← Older revision		Revision as of 09:41, 12 June 2007
Line 15:		Line 15:
	where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:		where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}		y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
Line 22:		Line 22:
	The inner region is given		The inner region is given

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	{\mu_t}_{inner} = \rho l^2 l \left[\left(		{\mu_t}_{inner} = \rho l^2 l \left[\left(
Line 32:		Line 32:
	where		where

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)		l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
Line 39:		Line 39:
	with the constant <math>\kappa = 0.4</math> and		with the constant <math>\kappa = 0.4</math> and

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.		A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.
Line 46:		Line 46:
	The outer region is given by:		The outer region is given by:

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),		{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),
Line 53:		Line 53:
	where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by		where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	\delta_v^* = \int_0^\delta (1-U/U_e)dy,		\delta_v^* = \int_0^\delta (1-U/U_e)dy,
Line 82:		Line 82:

	[[Category:Turbulence models]]		[[Category:Turbulence models]]
		+
		+	{{stub}}

Jola at 09:39, 12 June 2007

2007-06-12T09:39:17Z

← Older revision		Revision as of 09:39, 12 June 2007
Line 4:		Line 4:
	== Equations ==		== Equations ==

-	<table width="~~100~~%"><tr><td>	+	<table width="70%"><tr><td>
	:<math>		:<math>
	\mu_t =		\mu_t =
	\begin{cases}		\begin{cases}
	{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\		{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\
-	{\mu_t}_{outer} & \mbox{if} y > y_{crossover}	+	{\mu_t}_{outer} & \mbox{if } y > y_{crossover}
	\end{cases}		\end{cases}
	</math></td><td width="5%">(1)</td></tr></table>		</math></td><td width="5%">(1)</td></tr></table>

Jola at 09:36, 12 June 2007

2007-06-12T09:36:23Z

← Older revision		Revision as of 09:36, 12 June 2007
Line 1:		Line 1:
-	~~== Introduction ==~~	+	{{Turbulence modeling}}
-		+
	The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.		The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.

Littleandy: Category added

2006-10-04T07:51:31Z

Category added

← Older revision		Revision as of 07:51, 4 October 2006
Line 81:		Line 81:
	* {{reference-paper\|author=Smith, A.M.O. and Cebeci, T. \|year=1967\|title=Numerical solution of the turbulent boundary layer equations\|rest=Douglas aircraft division report DAC 33735}}		* {{reference-paper\|author=Smith, A.M.O. and Cebeci, T. \|year=1967\|title=Numerical solution of the turbulent boundary layer equations\|rest=Douglas aircraft division report DAC 33735}}
	* {{reference-book\|author=Wilcox, D.C. \|year=1998\|title=Turbulence Modeling for CFD\|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}		* {{reference-book\|author=Wilcox, D.C. \|year=1998\|title=Turbulence Modeling for CFD\|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}
		+
		+	[[Category:Turbulence models]]

Jasond: further clean-up, still needs work

2006-05-06T15:25:02Z

further clean-up, still needs work

← Older revision		Revision as of 15:25, 6 May 2006
Line 1:		Line 1:
	== Introduction ==		== Introduction ==

-	The Cebeci-Smith [[#References\|[~~Cebeci and~~ Smith (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects ~~(see below)~~. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.	+	The Cebeci-Smith [[#References\|[Smith and Cebeci (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.

	== Equations ==		== Equations ==
Line 21:		Line 21:
	</math></td><td width="5%">(2)</td></tr></table>		</math></td><td width="5%">(2)</td></tr></table>

-	The inner region is given ~~by the Prandtl - Van Driest formula:~~	+	The inner region is given

	<table width="100%"><tr><td>		<table width="100%"><tr><td>
	:<math>		:<math>
-	{\mu_t}_{inner} = \rho l^2 \left\| \~~Omega~~ \right\|	+	{\mu_t}_{inner} = \rho l^2 l \left[\left(
		+	\frac{\partial U}{\partial y}\right)^2 +
		+	\left(\frac{\partial V}{\partial x}\right)^2
		+	\right]^{1/2},
	</math></td><td width="5%">(3)</td></tr></table>		</math></td><td width="5%">(3)</td></tr></table>

Line 35:		Line 38:
	</math></td><td width="5%">(4)</td></tr></table>		</math></td><td width="5%">(4)</td></tr></table>

-	~~<table width="100%"><tr><td>~~	+	with the constant <math>\kappa = 0.4</math> and
-	:<math>	+
-	\kappa = 0.4~~, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}~~	+
-	</math>~~</td><td width="5%">(5)</td></tr></table>~~	+

	<table width="100%"><tr><td>		<table width="100%"><tr><td>
	:<math>		:<math>
-	\left\| \~~Omega~~ \~~right\| =~~ \~~sqrt{~~2 ~~\Omega_{ij~~} \~~Omega_~~{~~ij}~~}	+	A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.
	</math></td><td width="5%">(5)</td></tr></table>		</math></td><td width="5%">(5)</td></tr></table>
-
-	~~<table width="100%"><tr><td>~~
-	~~:<math>~~
-	~~\Omega_{ij} = \frac{1}{2}~~
-	~~\left(~~
-	~~\frac{\partial u_i}{\partial x_j} -~~
-	~~\frac{\partial u_j}{\partial x_i}~~
-	~~\right)~~
-	~~</math></td><td width="5%">(6)</td></tr></table>~~

	The outer region is given by:		The outer region is given by:
Line 59:		Line 50:
	:<math>		:<math>
	{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),		{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),
-	</math></td><td width="5%">(7)</td></tr></table>	+	</math></td><td width="5%">(6)</td></tr></table>

	where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by		where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by
Line 66:		Line 57:
	:<math>		:<math>
	\delta_v^* = \int_0^\delta (1-U/U_e)dy,		\delta_v^* = \int_0^\delta (1-U/U_e)dy,
-	</math></td><td width="5%">(8)</td></tr></table>	+	</math></td><td width="5%">(7)</td></tr></table>

	and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by		and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by
Line 74:		Line 65:
	F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6		F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6
	\right]^{-1}		\right]^{-1}
-	</math></td><td width="5%">(10)</td></tr></table>	+	</math></td><td width="5%">(8)</td></tr></table>


Line 88:		Line 79:
	== References ==		== References ==

-	*~~<b>~~Smith, A.M.O. and Cebeci, T.~~</b>~~ Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.	+	* {{reference-paper\|author=Smith, A.M.O. and Cebeci, T. \|year=1967\|title=Numerical solution of the turbulent boundary layer equations\|rest=Douglas aircraft division report DAC 33735}}
	* {{reference-book\|author=Wilcox, D.C. \|year=1998\|title=Turbulence Modeling for CFD\|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}		* {{reference-book\|author=Wilcox, D.C. \|year=1998\|title=Turbulence Modeling for CFD\|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}

Jasond: Copied from B-L model, still pretty rough

2006-05-05T18:22:54Z

Copied from B-L model, still pretty rough

← Older revision		Revision as of 18:22, 5 May 2006
Line 1:		Line 1:
-	== ~~References~~ ==	+	== Introduction ==

-	*<b>Smith~~, A.M.O. and~~ Cebeci, T.</b> ~~Numerical solution~~ of the ~~turbulent~~ boundary layer ~~equations~~, ~~Douglas aircraft division report DAC 33735~~.	+	The Cebeci-Smith [[#References\|[Cebeci and Smith (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects (see below). Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.

		+	== Equations ==

-	----	+	<table width="100%"><tr><td>
-	<i> ~~Return to [~~[~~Turbulence modeling]~~] </i>	+	:<math>
		+	\mu_t =
		+	\begin{cases}
		+	{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\
		+	{\mu_t}_{outer} & \mbox{if} y > y_{crossover}
		+	\end{cases}
		+	</math></td><td width="5%">(1)</td></tr></table>
		+
		+	where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
		+	</math></td><td width="5%">(2)</td></tr></table>
		+
		+	The inner region is given by the Prandtl - Van Driest formula:
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	{\mu_t}_{inner} = \rho l^2 \left\| \Omega \right\|
		+	</math></td><td width="5%">(3)</td></tr></table>
		+
		+	where
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
		+	</math></td><td width="5%">(4)</td></tr></table>
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	\kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}
		+	</math></td><td width="5%">(5)</td></tr></table>
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	\left\| \Omega \right\| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
		+	</math></td><td width="5%">(5)</td></tr></table>
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	\Omega_{ij} = \frac{1}{2}
		+	\left(
		+	\frac{\partial u_i}{\partial x_j} -
		+	\frac{\partial u_j}{\partial x_i}
		+	\right)
		+	</math></td><td width="5%">(6)</td></tr></table>
		+
		+	The outer region is given by:
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),
		+	</math></td><td width="5%">(7)</td></tr></table>
		+
		+	where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	\delta_v^* = \int_0^\delta (1-U/U_e)dy,
		+	</math></td><td width="5%">(8)</td></tr></table>
		+
		+	and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by
		+
		+	<table width="100%"><tr><td>
		+	:<math>
		+	F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6
		+	\right]^{-1}
		+	</math></td><td width="5%">(10)</td></tr></table>
		+
		+
		+	== Model variants ==
		+
		+
		+	== Performance, applicability and limitations ==
		+
		+
		+	== Implementation issues ==
		+
		+
		+	== References ==
		+
		+	*<b>Smith, A.M.O. and Cebeci, T.</b> Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.
		+	* {{reference-book\|author=Wilcox, D.C. \|year=1998\|title=Turbulence Modeling for CFD\|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}