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Extra info for Streamline Simulation: Theory and Practice

Sample text

39) Basic Governing Equations 23 x y Accumulation W z Out-Going Flux N Source R Inward Flux N Boundary of Volume A Volume V Fig. 1—General volume integral. Only the normal component of the flux contributes to the accumulation. Wi = ∫∫∫ Wi dV V ∫∫∫ dV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40) V This provides the general form of the finite-difference construction for the conservation equations. Wi n +1 n − Wi + n +1 ∫ n G dt ∫∫ J i • nˆ d A A ∫∫∫ dV = V n +1 ∫ dt∫∫∫ R dV ∫∫∫ dV .

Specifically, the mathematics of streamline simulation will utilize three important operations from vector calculus that are defined below for completeness. The readers can refer to any standard engineering mathematics text book for more details (Kreyszig, 2005). The gradient is a measure of the rate and direction of change in a scalar field. It is defined as ⎛ ⎞ ∇P = ⎜ iˆ ∂ + ˆj ∂ + kˆ ∂ ⎟ P = iˆ ∂P + ˆj ∂P + kˆ ∂P . ∂ ∂ ∂ x y z ∂x ∂y ∂z ⎝ ⎠ The gradient of a scalar field is a vector field and ∇P will be perpendicular to surfaces of constant P, and will point in the direction of maximum change of the pressure.

40) V This provides the general form of the finite-difference construction for the conservation equations. Wi n +1 n − Wi + n +1 ∫ n G dt ∫∫ J i • nˆ d A A ∫∫∫ dV = V n +1 ∫ dt∫∫∫ R dV ∫∫∫ dV . i n V . . . . . . . . . . . . 41) V Of the different terms, the most complicated to evaluate will be the integral of the flGux. To determine the change in the accumulation term, we must evaluate the normal component of the flux, J , on the boundary of the cell, and we must estimate how that flux varies with time.

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