Consider the transformation 1 ln y x � ξ Inverse transformation 1 � ξ e y x

# Consider the transformation 1 ln y x ? ξ inverse

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Consider the transformation: ) 1 ln( + = = y x η ξ Inverse transformation 1 = = η ξ e y x Example: Stretched (compressed ) grids The following derivatives are used in the transformation 1, 0 1 0, 1 x y x y y ξ ξ η η = = = = + 1, 0 0, x y x y e ξ ξ η η η = = = = ) 1 ln( + = = y x η ξ 1 = = η ξ e y x Example: Stretched (compressed ) grids The relation between increments y and η Therefore as η increases, y increases exponentially. Thus we can choose η constant and still have an exponential stretching of the grid in the y-direction. η η η η η η = = = e y d e dy e d dy Example: Stretched (compressed ) grids Transform the continuity equation Note there is relation between (x,y) plane and ( ξ , η ) plane Example: Stretched (compressed ) grids Substitute for the derivatives in Eq. (5.54) to get Eq. (5.57) is the continuity equation in the computational domain. Thus we have transformed the continuity equation from the physical space to the computational space. 