CFD Online URL
[Sponsors]
Home > Wiki > Linear Schemes - structured grids

Linear Schemes - structured grids

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
(Fromm - Fromm's Upwind Scheme)
 
Line 75: Line 75:
== Fromm - Fromm's Upwind Scheme ==
== Fromm - Fromm's Upwind Scheme ==
-
 
-
 
-
[[Image:NM_convectionschemes_struct_grids_Schemes_FROMM_Probe_01.jpg]]
 
== CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)  ==
== CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)  ==

Latest revision as of 01:11, 9 November 2005

Contents

Introduction

Linear schemes UDS, CDS and these

SOU - Second Order Upwind (also LUDS or UDS-2)

S.P.Vanka ({{{year}}}), "Second-order upwind differencing ina recirculating flow", AIAA J., 25, 1435-1441.

R.F.Warming and R.M. Beam (1976), "Upwind second order difference schemes and applications in aerodynamics flows", AIAA J. 14 (1976) 1241-1249.

Skew - Upwind

G.D.Raithby , Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)

QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS)

B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98

Usual variables


 
	\phi_{w}= \frac{3}{8}\phi_{P}+ \frac{3}{4}\phi_{W} - \frac{1}{8}\phi_{WW}
(2)
 
	\phi_{f}= \frac{3}{8}\phi_{D}+ \frac{3}{4}\phi_{C} - \frac{1}{8}\phi_{U}
(2)


Normalised variables (uniform grid)


 
	\hat{\phi_{f}}= \frac{3}{8} + \frac{3}{4}\hat{\phi_{C}}
(2)


Normalised variables (non-uniform grid)

 
\begin{matrix}
\hat{\phi_{w}} & =  \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{W}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right]  \right\} U^{+}_{w} + \\
+ &	\left\{ C_{2} \left( 1 + C_{3} \right) \hat{\phi_{P}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right]  \right\} U^{-}_{w}
\end{matrix}
(2)
 
\begin{matrix}
\hat{\phi_{f}} & =  \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{C}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right]  \right\} U^{+}_{f} + \\
+ &	\left\{ C_{2} \left( 1 + C_{3} \right) \hat{\phi_{D}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right]  \right\} U^{-}_{f}
\end{matrix}
(2)

LUS - Linear Upwind Scheme

H.C.Price, R.S. Varga and J.E.Warren (1966), "Application of oscillation matrices to diffusion-convection equations", Journal Math. and Phys., Vol. 45, p.301, (1966).

 
   \phi_{f}= \phi_{C} + 0.5 \left( \phi_{C} - \phi_{U} \right)
(1)

Fromm - Fromm's Upwind Scheme

CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)

In CUDS (UDS-4) for interpolation of function is used three upwind nodes and one node downstream.

usual variables

 
	\phi_{w}=\frac{1}{3}\phi_{P} + \frac{5}{6}\phi_{W} + \frac{1}{6}\phi_{WW}
(2)

normalised variables (uniform grids)

 
	\hat{\phi_{w}}=\frac{1}{3} + \frac{5}{6}\hat{\phi_{W}}
(2)

R.K. Aragval

A third-order-accurate upwind scheme for Navier-Stokes solution at high Reynolds numbers

Paper No. AIAA-81-0112, AIAA 19th Aerospace Science Meeting, St. Louis, 1982.

CUI - Cubic Upwind Interpolation

B.P. Leonard

A survey of finite differences of opinion on numerical muddling of incompressible defective confusion equation

paper in ASME, Applied Mechanics Division, Winter Annual Meeting, 1979




Return to Numerical Methods

Return to Approximation Schemes for convective term - structured grids

My wiki