Cebeci-Smith model

(Difference between revisions)
 Revision as of 02:00, 24 November 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 18:22, 5 May 2006 (view source)Jasond (Talk | contribs) (Copied from B-L model, still pretty rough)Newer edit → Line 1: Line 1: - == References == + == Introduction == - *Smith, A.M.O. and Cebeci, T. 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, $\mu_t, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace applications. Like the [[Baldwin-Lomax model]], this model is not suitable for cases with large separated regions and significant curvature/rotation effects (see below). Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge. + == Equations == - ---- + - Return to [[Turbulence modeling]] + :[itex] + \mu_t = + \begin{cases} + {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ + {\mu_t}_{outer} & \mbox{if} y > y_{crossover} + \end{cases} +$(1)
+ + where $y_{crossover}$ is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$: + +
+ :$+ y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer} +$(2)
+ + The inner region is given by the Prandtl - Van Driest formula: + +
+ :$+ {\mu_t}_{inner} = \rho l^2 \left| \Omega \right| +$(3)
+ + where + +
+ :$+ l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right) +$(4)
+ +
+ :$+ \kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2} + (5) + + + :[itex] + \left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}} +$(5)
+ +
+ :$+ \Omega_{ij} = \frac{1}{2} + \left( + \frac{\partial u_i}{\partial x_j} - + \frac{\partial u_j}{\partial x_i} + \right) +$(6)
+ + The outer region is given by: + +
+ :$+ {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), +$(7)
+ + where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by + +
+ :$+ \delta_v^* = \int_0^\delta (1-U/U_e)dy, +$(8)
+ + and $F_{KLEB}$ is the Klebanoff intermittency function given by + +
+ :[itex] + F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 + \right]^{-1} + (10)
+ + + == Model variants == + + + == Performance, applicability and limitations == + + + == Implementation issues == + + + == References == + + *Smith, A.M.O. and Cebeci, T. 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.}}

Introduction

The Cebeci-Smith [Cebeci and Smith (1967)] is a two-layer algebraic 0-equation model which gives the eddy viscosity, $\mu_t$, 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

 $\mu_t = \begin{cases} {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ {\mu_t}_{outer} & \mbox{if} y > y_{crossover} \end{cases}$ (1)

where $y_{crossover}$ is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$:

 $y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}$ (2)

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

 ${\mu_t}_{inner} = \rho l^2 \left| \Omega \right|$ (3)

where

 $l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)$ (4)
 $\kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}$ (5)
 $\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}$ (5)
 $\Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$ (6)

The outer region is given by:

 ${\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),$ (7)

where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by

 $\delta_v^* = \int_0^\delta (1-U/U_e)dy,$ (8)

and $F_{KLEB}$ is the Klebanoff intermittency function given by

 $F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 \right]^{-1}$ (10)

References

• Smith, A.M.O. and Cebeci, T. Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.
• Wilcox, D.C. (1998), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..